Peak capacity estimation in isocratic elution

Journal of Chromatography A, 1205 (2008) 78–89

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Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma

Peak capacity estimation in isocratic elution S. Pous-Torres, J.J. Baeza-Baeza ∗ , J.R. Torres-Lapasió, M.C. García-Álvarez-Coque Departament de Química Analítica, Universitat de València, c/Dr. Moliner 50, 46100 Burjassot, Spain

a r t i c l e

i n f o

Article history: Received 14 May 2008 Received in revised form 23 July 2008 Accepted 31 July 2008 Available online 5 August 2008 Keywords: Reversed-phase liquid chromatography Hydro-organic mixtures Micellar eluents Peak Capacity Giddings’ approach Grushka’s approach

a b s t r a c t Peak capacity (i.e. maximal number of resolved peaks that fit in a chromatographic window) is a theoretical concept with growing interest, but based on a situation rarely met in practice. Real chromatograms tend to have uneven distributions, with overlapped peaks and large gaps. The number of resolved compounds should, therefore, be known from estimations. Several equations have been reported for this purpose based on three perspectives, namely, the intuitive approach (peak capacity as the size of the retention time window measured in peak width units), which assumes peaks with the same width, and the outlines of Giddings and Grushka, which consider changes in peak width with retention time. In this work, the peak capacity concept is discussed and three new approaches are derived based on realistic descriptions of peak shape. The first one is based on the Grushka’s approach and considers the contributions of column and extra-column peak variances. The second one relies on Giddings’ and assumes asymmetrical peaks where left and right peak half-widths depend linearly on retention time. The third equation, based on the intuitive approach, uses a mean peak width obtained by integration, instead of a mean value from several representative peaks. The accuracy of the classical Giddings’ approach for ideal peaks, a modification of the Grushka’s approach that considers variation of peak width at half-height, and the three new approaches were checked on combined chromatograms built by adding real peaks. The results demonstrate that the change in efficiency (and not in skewness) is the relevant factor, at least in the studied examples. Also, peak width should be measured at low peak height ratios (i.e. 10%) to better account peak deformation. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Analysts qualify a chromatographic system based on its capability to achieve good separation in reasonable run time, for which not only retention times, but also peak widths should be taken into account. Accordingly, considerable work has been invested on the factors controlling peak retention and width in chromatography. In recent time, peak capacity has become a useful parameter to evaluate the overall separation potential of a chromatographic system. Initially developed for isocratic elution, peak capacity has called attention especially in uni- and two-dimensional systems with gradient elution, especially for samples that contain many components where achieving complete resolution is problematic. The recent reviews written by Neue [1] and the research groups of Carr and Rutan [2] are good examples of the increasing interest in peak capacity estimation. The peak capacity concept is based on the fact that the peaks from all chromatographic columns have a finite width dictated by the plate number, N, and consequently, only a limited number of

∗ Corresponding author. Tel.: +34 96 354 3184; fax: +34 96 354 4436. E-mail address: [email protected] (J.J. Baeza-Baeza). 0021-9673/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2008.07.088

peaks can fit into the accessible time range. Peak capacity is defined as the maximal number of “resolvable” or “resolved” peaks (i.e. adjacent peaks located at exactly the right distance to yield enough resolution) that can be separated in a certain time window, under prescribed experimental conditions [3]. This assessment has the interest of considering the entire chromatographic space, using the information about the column efficiency. It should be remarked, however, that peak capacity is only accessible through calculation. It is, therefore, a theoretical concept. The maximal number of resolved peaks in the sense considered in its definition exceeds what can be found in a real chromatogram. Chromatographic peaks are rarely regularly spaced (as found, for example, for oligomers of fatty alcohol ethoxylates [4]): some components share the same space in the chromatogram and the gaps are quite abundant. The chromatographic window required to separate all sample components at a given probability level was first demonstrated to increase with the squared number of components [5]. More recently, an experimental verification and modification of this idea has been published [1]. Finding ways to estimate accurately peak capacity in any situation deserves still some attention. The approaches reported in the literature assume ideal peaks, and peak widths or efficiencies are often considered unchanged with retention time [6–13]. This work

S. Pous-Torres et al. / J. Chromatogr. A 1205 (2008) 78–89

discusses in detail the reported approaches, and suggests new perspectives for peak capacity estimation where changes in peak width and asymmetry with retention time are considered. The accuracy of the new approaches was checked using chromatographic data of several sets of compounds eluted with isocratic hydro-organic and micellar reversed-phase liquid chromatography.

2.2. Assumption of invariable peak efficiency 2.2.1. Giddings’ approach Giddings [3] quantified the peak capacity based on the following: tR,i+1 − tR,i =

2. Approaches to estimate peak capacity In Sections 2.1, 2.2 and 2.3.1, some reported approaches for peak capacity estimation in isocratic elution are described. Knowledge of their basis is convenient to understand the meaning of the new approaches introduced in Section 2.3.

79

4i+1 + 4i wi+1 + wi = 2 2

which indicates that the distance between two adjacent peaks that touch each other is not greater than the mean of the two peak widths. Peak width was taken again as 4 for granting resolution (RS = 1.0). From tR  = √ = m tR N

(6)

2.1. Assumption of invariable peak widths: the intuitive approach

the following was obtained:

Peak capacity measures the number of chromatographic peaks touching each other that can fit in a certain time window. The most simple and intuitive way of estimating it is as the size of the retention time window measured in peak width units, assuming that all peaks have the same width:

tR,i+1 − tR,i = ˛

Pc = 1 +

t w

(1)

where t is the time window measured between the maxima of the two extreme peaks, and w is the mean peak width, which is usually taken as 4 (i.e. the width at 13.4% of peak height), although at this level the separation is not complete. This assumption implicitly considers RS = 1, where RS is the classical resolution: RS =

tR,2 − tR,1 (w1 + w2 )/2

t Pc = 1 + RS w

(3)

The “1” term is added to take into account the left and right halfwidths of the first and last eluting peaks, respectively. The lower bound of t can be the peak of an unretained compound (i.e. eluted at the dead time) or an arbitrary peak at the beginning of the chromatogram (frequently, the first eluting peak), and the upper bound is the retention time of a selected peak (frequently the last eluted). For gradient elution, t is often taken as tG –tD –t0 (gradient run time, dwell time and dead time, respectively) [1]. Peak capacity in the gradient mode is larger than in the isocratic mode owing to the peak compression effect. Eq. (1) ignores the changes in peak width with retention time. In some chromatographic modes under gradient elution, peak width is indeed rather uniform throughout the chromatogram (especially for small t) [14], and one can select any peak to measure the width, but in other cases this may easily increase with retention. Since measuring the widths of multiple peaks in a chromatogram is tedious, very often peak capacity is estimated by averaging only a few width measurements. Dolan et al. [14] related the definitions of Pc and RS (Eq. (2)), and proposed a modified peak capacity estimation, which was called the “sample peak capacity”: Pc∗∗ =

t w

For sufficiently high peak capacities: Pc ≈ Pc∗∗ .

(4)

tR,i+1 + tR,i 2

(7)

where 4 ˛= √ N

(8)

From Eq. (7), the ratio of retention times for any two adjacent peaks will be: tR,i+1 1 + ˛/2 = tR,i 1 − ˛/2

(9)

and for the first and last eluting peaks: tR,n = tR,1

 

(2)

applied to neighbouring peak pairs with retention times tR,1 and tR,2 , and widths w1 and w2 , which are also usually measured as 4. For other RS values, Eq. (1) can be expressed as:

(5)

=

tR,2 tR,1



1 + ˛/2 1 − ˛/2

tR,3 tR,2





...



tR,n



tR,n−1

1 + ˛/2 1 − ˛/2



 ...

1 + ˛/2 1 − ˛/2



 =

1 + ˛/2 1 − ˛/2

n−1 (10)

where Pc = n. In logarithmic form: ln

tR,n 1 + ˛/2 = (Pc − 1) ln tR,1 1 − ˛/2

(11)

and finally: Pc = 1 +

ln(tR,n /tR,1 ) ln((1 + ˛/2)/(1 − ˛/2))

(12)

In most practical situations, N is large enough (>103 ), and consequently, ˛ is sufficiently small (Eq. (8)). Based on this observation, Giddings expanded the denominator in Eq. (12) as



ln

1 + ˛/2 1 − ˛/2



=˛+

1 3 ˛ + ... 3

and discarded all terms, except the first one to yield: √ tR,n tR,n N 1 =1+ ln ln Pc = 1 + ˛ 4 tR,1 tR,1

(13)

(14)

This simplification implies an error slightly larger than 1% for N = 500 (a pessimistic figure in chromatography). Peak capacity was also expressed as a function of the retention factors for the first (k1 ) and last (kn ) eluted peaks as √ 1 + kn N ln Pc = 1 + (15) 4 1 + k1 being: ki =

tR,i − t0 t0

(16)

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Eqs. (14) and (15) consider peak widening with retention time through the efficiency term, which is affected by column diameter and length, particle size, flow rate, and gradient slope (in gradient elution). Giddings assumed that the column provided equal efficiency for all eluted solutes [3], so peak widths would increase proportionally to retention times following the ideal behaviour (Eq. (6)). Also, tR,1 was not necessarily the retention time of an unretained compound. 2.2.2. Grushka’s approach Grushka [6] outlined an infinitesimal treatment of the definition of peak capacity (see also Eqs. (1) and (6)): √ dt N dt dt dn = = = (17) w 4 4 t which is integrated as follows:  n √  tR,n N dt dn = n − 1 = Pc − 1 = 4 t 1 t

2.3. Efficiency dependence with retention time As commented, Giddings and Grushka assumed that N is the same for all solutes. However, N often tends to increase with retention, and invariable N assumption for solutes having different retention may result in wrong peak capacity estimation. We will consider again the three above approaches (the intuitive approach, and the approaches proposed by Giddings and Grushka), assuming changes in N with retention. We will start with some reported equations, which are based on the approach proposed by Grushka, and then we will introduce new perspectives based on Grushka’s, Giddings’ and the intuitive approach. 2.3.1. Grushka’s approach At least three equations based on the Grushka’s approach have been reported that consider the dependence of N with retention. In one of the proposed modifications of Grushka’s, N was assumed to follow a polynomial dependence with the retention factor [15]:  tR,n √ 1 N Pc = 1 + dt 4 t t 1 4

= 1+

k



(N∞ + b/k + c/k2 + . . .) 1+k

0

dk

(20)

For sufficiently large k values, N → N∞ . A linear relationship was later assumed between the peak width at half-height (w1/2 ) and the retention time [16]: w1/2 = atR − b

(21)

where a and b are fitting parameters, with b reflecting any extracolumn contribution. Considering that:

 N = 5.54

tR w1/2

1 Pc = 1 + 4

k =

Eq. (19) implies that peak capacity increases with the retention factor. The increment rate, however, decreases with k.

2

tR,n

tR,1

1 dt atR − b (23)



kn

0

1 dk k

(24)



t 1+k = (1 + k) = √ t0 N

H L

(25)

L being the column length and H the plate height. The following results: √  kn 1 1 L dk (26) Pc = 1 +  4 0 (1 + k) H(k) Eq. (26) was integrated by the authors piecewise, assuming that H was approximately the same for two adjacent peaks [17]. In a chromatogram with n components, this yielded: √  n  ki L Pc = 1 + 4 k i=1

= 1+

1



i−1

1 dk (1 + k) Hi

√  n 1 + ki L 1  ln 4 1 + ki−1 H

(27)

i

i=1

The authors finally made a simplification assuming peaks with a mean plate height: √ L 1 Pc = 1 + (28)  ln (1 + kn ) 4 H which coincides with Eq. (19). In fact, multiple ways to include changes in efficiency or peak width with retention time in the Grushka’s integral (Eqs. (18) or (24)) are possible. Ideally, the dependence functions should have physical meaning, but at the same time they should be simple and integrable. Next, an approach meeting these conditions is given. It should be first considered that the overall performance of a chromatographic system is given by the performance of the column itself and the instrumental contribution to band broadening: 2 2 2 tot = col + ext

(29)

2 is the observed peak variance,  2 the column variance, where tot col 2 accounts for the extra-column contributions (e.g. conand ext necting tubes, injection plug, detector cell) [18,19]. The change in efficiency with retention can be interpreted as a relative decrease in importance of the extra-column contribution to band broadening. In the absence of particular interactions between the solutes and stationary phase, the N value for a column (Ncol ) can be assumed to be independent of the retention time. From Eq. (6):



(22)



As tR increases, the term b becomes negligible, and Eq. (23) reduces to Eq. (14), as reported by Giddings and Grushka. It should be noted that this approach can be easily adapted to other peak height ratios. Another equation was obtained by integration of Eq. (17) expressed as a function of the standard deviation, considering retention factors instead of retention times [17]:

(18)

This gives rise directly to the Giddings’ solution (Eq. (14)). However, Grushka took tR,1 as the dead time, which simplifies Eq. (15): √ N Pc = 1 + (19) ln (1 + kn ) 4

0

R,1

√ atR,n − b 5.54 ln = 1+ 4a atR,1 − b

where:

R,1



the following was obtained:  tR,n √ √ 1 N 5.54 dt = 1 + Pc = 1 + 4 t t 4

2 tot

=

m2 tR2

2 + ext

=

m2

tR2

+

2 ext

m2



(30)

S. Pous-Torres et al. / J. Chromatogr. A 1205 (2008) 78–89

and from Eq. (17): Pc = 1 +

1 4m



tR,n

tR,1

⎛

1 = 1+ ln 4m

⎝



tR2

tR,n + tR,1 +

Giddings assumed that chromatographic peaks were perfect Gaussians, with widths w = 4 (RS = 1, Eq. (5)). In order to consider a more general situation, Eq. (7) can be rewritten in standardised units (z):

dtR 2 /m2 ) + (ext 



2 + ( 2 /m2 ) tR,n  ext



⎞ ⎠

(31)

2 + ( 2 /m2 ) tR,1  ext

2 , and therefore, This equation requires knowledge of m and ext of the total peak variance (Eq. (30)). This is not straightforward for asymmetrical real peaks [20]. We suggest here three different approaches for this calculation. The first approach considers that the equivalent peak variance for asymmetrical peaks can be calculated from the second order moment, according to the approximation given by Foley and Dorsey [20]:

M2 =

2 w10% 2

1.764(B/A) − 11.15(B/A) + 28

(32)

2 ) are measured at 10% peak height. Eq. (32) where the widths (w10% was reported to yield errors <1.5% for asymmetries (B/A ratio, being A and B the left and right half-widths, respectively, measured at 10% peak height) in the range 1.00–2.76. For the other two approaches:

w 2

2 tot =

1 4.6

2 = tot

A2 + B2 2 × 4.6

2

=

(A + B)2 4 × 4.6

81

tR,i+1 − tR,i = 2zm

Pc = 1 + and Pc = 1 +

(35)

B = mB (tR − t0 ) + B0 = mB t + B0

(36)

where A0 and B0 (i.e. the half-widths of a peak eluting at the dead time, t0 ) account for the extra-column contributions, which as commented above are more significant for the narrower peaks at the shorter retention times. For large windows, the curvature cannot be neglected anymore, and parabolic relationships are required [21]. (iii) The behaviour described in Eq. (30) cannot be incorporated in the Giddings’ approach. (iv) Finally, by splitting the peak width in linear terms for the two half-widths, the asymmetry remains explicit in the modified Giddings’ expression.

√ tR,n 1 tn + t0 N ln =1+ ln 2zm tR,1 2z t1 + t0

(39)

(40)

which implicitly assumes that A and B are measured from the peak centre to the peak inflection points. The following is then obtained:

Thus

A = mA (tR − t0 ) + A0 = mA t + A0

(38)

ti+1 − ti = z(Ai+1 + Bi ) = z(mA ti+1 + A0 + mB ti + B0 )

(34)

(i) Linear relationships between peak width (i.e. standard deviation) and retention time with null intercept in Eq. (6) hold only for ideal chromatographic peaks. (ii) For convenience, Eq. (30) can be approximated in moderate time windows to a linear dependence between peak width (or standard deviation) and retention time (Eq. (21)). This allows also relating left (A) and right (B) half-widths individually to the retention time:

ln(tR,n /tR,1 ) ln((1 + zm )/(1 − zm ))

Let’s consider now that instead of an ideal peak, each peak has particular half-widths, which vary with retention time according to Eqs. (35) and (36). Similarly to Eq. (37), we can write:

ti+1 =

2.3.2. Modified Giddings’ approach for non-Gaussian peaks In the approaches described above, ideal elution profiles were assumed. Several phenomena may yield, however, asymmetrical chromatographic peaks with larger widths than expected. We reconsider next the approach proposed by Giddings, where changes in width and asymmetry of chromatographic peaks are assumed (i.e. modified Giddings’ approach). Before proceeding, one should consider the following:

(37)

Giddings adopted the value z = 2 (˛ = 2zm = 4m ). Eq. (37) allows / 1). Accordingly, Eqs. (12) and (14) can any RS value (i.e. RS = 1 or = be reformulated as

(33)

The factor 4.6 has been included to account for the measurement of A and B at 10% peak height.

tR,i+1 + tR,i 2

1 + zmB A0 + B0 t +z = ıti + 0 1 − zmA i 1 − zmA

(41)

t2 = ıt1 + 0

(42)

t3 = ıt2 + 0 = ı2 t1 + ı0 + 0

(43)

t4 = ıt3 + 0 = ı3 t1 + ı2 0 + ı0 + 0

(44)

and for any peak: ti = ıi−1 t1 +

i−2 

ıj 0 = ıi−1 t1 +

j=0

ıi−1 − 1 0 ı−1

(45)

For convenience, Eq. (45) will be expressed as ti = ıi−1 t1 + (ıi−1 − 1)T0

(46)

From Eqs. (41) and (45): T0 =

0 z((A0 + B0 )/(1 − zmA )) A0 + B0 = = mA + mB ı−1 ((1 + zmB )/(1 − zmA )) − 1

(47)

Therefore, T0 is the time for an ideal peak with width (A0 + B0 ). Finally Pc = 1 +

ln((tn + T0 )/(t1 + T0 )) ln((tn + T0 )/(t1 + T0 )) =1+ ln ı ln((1 + zmB )/(1 − zmA )) (48)

Giddings arrived to a similar equation (Eq. (12), see also Eq. (38)), but decided to substitute the logarithmic term in the denominator to yield the simpler Eq. (14). Eq. (12) is indeed already simple, thus, in our opinion this operation can be skipped. However, for comparison purposes we expanded the denominator in Eq. (48) as follows: ln

m2 − m2B 1 + zmB = z(mA + mB ) + z 2 A 1 − zmA 2 + z3

(mA + mB )(m2A + m2B − mA mB ) 3

+ ...

(49)

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S. Pous-Torres et al. / J. Chromatogr. A 1205 (2008) 78–89

and discarded all terms, except the first one to give: Pc = 1 +

1 tn + T0 ln t1 + T0 z(mA + mB )

(50)

that can be compared with Eqs. (14) or (39). For an ideal symmetrical peak, mA = mB = m , and A0 = B0 = m t0 . From Eq. (47): T0 =

2m t0 = t0 2m

(51)

and Eq. (50) yields Eq. (39). 2.3.3. The intuitive approach including efficiency changes The simple intuitive approach (Eq. (3)) also allows changes in efficiency with retention time. For this purpose, instead of an average peak width determined from several representative peaks, mean left and right half-widths were used: Pc = 1 +

tn − t1

(52)

z(A + B)

which can be calculated by integration as follows:

 tn

A=

t1

A(t) dt

 tn t1

dt

 tn

=

t1

A(t) dt

(53)

tn − t1

Assuming again a linear relationship between left half-width and time (Eq. (35)), integration of Eq. (53) yields: A=

(mA /2)(tn2 − t12 ) + A0 (tn − t1 ) tn − t1

=

mA (tn + t1 ) + A0 2

(54)

and similarly for the mean right half-width from Eq. (36): B=

mB (tn + t1 ) + B0 2

(55)

By substitution of Eqs. (54) and (55) in Eq. (52): Pc = 1 +

tn − t1 z(((mA + mB )/2)(tn + t1 ) + (A0 + B0 ))

(56)

In contrast with Giddings’, this approach admits relationships between width and time of higher order. Thus, for a parabolic dependence between A and tR , Eq. (53) will yield: A¯ =

(aA /3)(tn3 − t13 ) + (mA /2)(tn2 − t12 ) + A0 (tn − t1 ) tn − t1

(57)

3. Experimental Chromatographic data (i.e. retention times, heights, left and right half-widths) gathered in our laboratory along several years were used to assess the accuracy of peak capacity estimation. This database includes information from more than one thousand peaks, corresponding to the following compound sets: (i) Sulfonamides: sulfacetamide, sulfachloropyridazine, sulfadiazine, sulfadimethoxine, sulfaguanidine, sulfamerazine, sulfamethizole, sulfamethoxazole, sulfamonomethoxine, sulfanilamide, sulfaquinoxaline and sulfisoxazole (Sigma, St. Louis, MO, USA), and sulfamethazine from (Aldrich, Milwaukee, WI, USA). (ii) Steroids: clostebol acetate, dehydrotestosterone, metenolone enanthate and methyltestosterone (Sigma), dydrogesterone (Kalifarma, Barcelona, Spain), medroxyprogesterone acetate (Cusí, Barcelona), nandrolone (Fher, Barcelona), nandrolone decanoate (Organón, Barcelona), progesterone (Seid, Barcelona), testosterone, testosterone enanthate and testosterone propionate (Schering, Madrid, Spain). (iii) ␤-Blockers: acebutolol (Italfarmaco, Alcobendas, Madrid), alprenolol, pindolol, sotalol (Sigma), atenolol (Zeneca Farma,

Madrid), bisoprolol, propranolol, practolol (ICI-Farma, Madrid), carteolol (Miquel-Otsuka, Barcelona), celiprolol (Rhône-Poulenc Rorer, Alcorcón, Madrid), esmolol (Polfa, Starogard, Poland), labetalol (Glaxo, Tres Cantos, Madrid), metoprolol, oxprenolol (Ciba-Geigy, Barcelona), nadolol (Squibb, Esplugues de Llobregat, Barcelona) and timolol (Merck, Sharp & Dohme, Madrid). (iv) Tricyclic antidepressants: amitryptiline, clomipramine, doxepin, imipramine, maprotiline, nortryptiline and trimipramine (Sigma). (v) Diuretics: althiazide, benzthiazide, bumetanide, chlorothiazide, furosemide, hydrochlorothiazide, probenecid, triamterene, trichloromethiazide (Sigma), amiloride (ICI-Farma), bendroflumethiazide (Davur, Madrid), chlorthalidone (CibaGeigy), ethacrynic acid (Merck, Sharp & Dohme), piretanide (Cusí), spironolactone (Searle, Madrid) and xipamide (Lacer, Barcelona). In the hydro-organic mode, the mobile phases contained acetonitrile, methanol (Scharlab, Barcelona, Spain) and triethylamine (TEA, Fluka, Buchs, Switzerland), and in the micellar mode, sodium dodecyl sulfate (99% purity, Merck, Darmstad, Germany), and acetonitrile, propanol, and pentanol (Scharlab). The pH was buffered in the range 3–7 with citric acid monohydrate or disodium hydrogenphosphate, and NaOH or HCl (Panreac, Barcelona). Chromatographic data were obtained using several HPLC systems from Agilent (Series 1050 and 1100, Palo Alto, CA, USA), equipped with isocratic pumps, autosamplers, UV–visible detectors and temperature controllers. The chromatographic columns (125 × 4.6 mm and 5 ␮m particle size) were the following: unendcapped Spherisorb ODS-2 (Scharlab), Hypersil C18 (Agilent, Waldbronn, Germany), Kromasil C18 (Análisis Vínicos, Ciudad Real, Spain), XTerra MS C18 (Waters, MA, USA), and Zorbax SB C18 (Agilent). 4. Results and discussion Eq. (14) or the more general Eq. (39) are suited to measure peak capacity for symmetrical peaks (mA = mB ), whenever efficiency remains constant throughout the chromatogram. Since usually this is not the case, peak capacity tends to be overestimated, as demonstrated below. We propose here new equations (Eqs. (31), (48) and (56)), which allow efficiency changes in peak capacity estimation. Eq. (48) allows also different values for the left and right peak halfwidths (mA = / mB ). 4.1. Dependence of half-widths with retention time For ideal chromatographic peaks, the standard deviation is linearly related to the retention time with null intercept (Eq. (6)). For skewed peaks, left and right half-widths depend on retention time with residual values at t0 . This dependence is approximately linear (Eqs. (35) and (36)). As will be shown, the linear relationships hold satisfactorily, at least for usual time windows. In this work, half-widths were measured at 10% peak height. From our experience, measurement at this height ratio is convenient and peak asymmetries are adequately accounted. In RPLC with hydro-organic mobile phases, using the same solute set with all solutes exhibiting approximately the same intrinsic efficiency and tailing, the parameters mA and mB are characteristic of a given column/solvent system at fixed temperature. Accordingly, Eqs. (35) and (36) can be fitted either using the retention data of several compounds with different polarities eluted with the same mobile phase, or a selected compound

S. Pous-Torres et al. / J. Chromatogr. A 1205 (2008) 78–89

83

Fig. 1. Dependence of left (A, ) and right (B, 䊉) half-widths with retention time for: (a) a set of 10 diuretics and ␤-blockers eluted with 35% (v/v) acetonitrile, (b) xipamide eluted with 25–50% acetonitrile, (c) 13 diuretics eluted with 0.15 M SDS/20% acetonitrile, and (d) xipamide eluted with 0.05–0.15 M SDS/10–20% acetonitrile. In all instances, the mobile pH was 3.

eluted with mobile phases of diverse composition (i.e. solvent content). Fig. 1a depicts the correlation of A and B with retention time for a set of 10 diuretics and ␤-blockers (sorted according to their elution from less to more retained): oxprenolol, chlorthalidone, bendroflumethiazide, propranolol, alprenolol, furosemide, benzothiazide, xipamide, bumetanide and probenecid, eluted with a mobile phase of 35% (v/v) acetonitrile. Fig. 1b shows the behaviour for xipamide, eluted with mobile phases of acetonitrile in the range 25–50%. In both cases, a practically linear dependence was achieved, especially for the left half-width (A). One reason of the better linearity for A could be the higher accuracy in detecting the point at 10% peak height in the fronting part of the peak. As can be observed, the peaks are tailing (B > A), although peak symmetry is enhanced with retention. The organic solvent is adsorbed on the stationary phase, which affects solute partitioning. Accordingly, some changes in the peak parameters were observed with mobile phase composition. Thus, for the set of diuretics and ␤-blockers eluted with 30 and 45% acetonitrile: mA = 0.0206 and 0.0182, mB = 0.0183 and 0.0154,

A0 = 0.0464 and 0.0478, and B0 = 0.104 and 0.102, respectively, with R2 > 0.996. Such a result suggests that measurement of peak shape parameters should be preferably done with a set of compounds eluted with a unique mobile phase. The behaviour is more complex for hydro-organic mixtures containing a micellised surfactant (i.e. micellar RPLC, or MLC). Fig. 1c shows the dependence between left and right halfwidths and retention time for a set of 13 diuretics, some of them included in the previous set (sorted according to their elution order): trichloromethiazide, chlorthalidone, althiazide, furosemide, benzthiazide, bendroflumethiazide, piretanide, amiloride, bumetanide, probenecid, xipamide, ethacrynic acid and triamterene. The compounds were eluted with 0.15 M sodium dodecyl sulfate (SDS)/20% (v/v) acetonitrile. Satisfactory linearity was achieved again. However, the situation changed when a selected compound (or a set of compounds) was eluted with micellar mobile phases at diverse concentrations of SDS and acetonitrile. Fig. 1d shows the correlation between half-widths and retention time for xipamide eluted with several mobile phases in the ranges 0.05–0.15 M SDS and 10–20% (v/v) acetonitrile. The correlation is

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significantly poorer with respect to the use of a unique mobile phase composition. In MLC, the relationships between peak shape parameters and retention time (Eqs. (35) and (36)) depend on both surfactant and organic solvent [22]. In this chromatographic mode, the surfactant covers the stationary phase. Silanol and alkylbenzene groups in conventional RPLC columns are thus screened, giving rise to new interactions between the solutes and the polar and non-polar groups of the surfactant. Depending on the charge of solute and surfactant (i.e. neutral, cationic or anionic), these interactions can be solely hydrophobic or also electrostatic (repulsion or attraction). The addition of organic solvent affects the nature of micelles in the mobile phase and reduces the thickness of the surfactant layer on the stationary phase. This modifies the retention capability and the adsorption/desorption kinetics [23]. There is an extensive demonstration of the reduction in efficiency and symmetry with increased concentration of surfactant in the mobile phase, and the enhancement of peak shape parameters upon the addition of an organic solvent, especially in the case of acetonitrile [24,25]. Thus, for example, for the set of diuretics eluted with 0.15 M SDS (without organic solvent) and 0.15 M SDS/20% acetonitrile: mA = 0.0417 and 0.0268, mB = 0.0467 and 0.0248, A0 = 0.0686 and 0.0641, and B0 = 0.110 and 0.128, respectively. Therefore, the scattering observed in Fig. 1d is a consequence of the change in the surfactant/organic solvent ratio that modifies the thickness of the surfactant layer on the stationary phase. Indeed, it is possible to find two mobile phases giving rise to the same retention, but different coverage of surfactant, and consequently, different efficiency. 4.2. Simulation of combined chromatograms with asymmetrical peaks The equations in Section 2 offer estimations of the number of peaks touching each other that fit in a given time window. We were concerned about the accuracy of these estimations, but real chromatograms illustrating the peak capacity definition are not easy to find. We thought then in an alternative to check the number of peaks that really fit in a given time window: the simulation of chromatograms with resolved peaks. In order to increase realism, we built combined chromatograms by adding real peaks obtained with an unendcapped Spherisorb ODS-2 column, whose descriptors (i.e. retention time, peak height, and left and right half-widths) were in the database introduced in Section 3. The peaks were obtained with mobile phases of different nature (hydro-organic or micellar) and were mainly asymmetrical with B/A > 1.2, exceeding in some cases B/A > 2.5.

Fig. 2. Section of a combined chromatogram built with real peaks eluted with acetonitrile-water mixtures, using an unendcapped Spherisorb ODS-2.

In contrast with previous estimations of peak capacity that assumed RS = 1.0 (i.e. w = 4), we decided to increase the separation between consecutive peaks to guarantee near baseline resolution, adopting RS = 1.5 (i.e. w = 6). The reason is that nonGaussian chromatographic peaks undergo a pseudo-exponential decrease close to the baseline that increases the probability of overlapping between adjacent peaks. For ideal peaks and the non-ideal peaks considered in this work, the overlapping amounted 4.6% and 5.5–7% for RS = 1.0, and 0.3% and 0.6–2.5% for RS = 1.5, respectively. Thus, z = 3 instead of z = 2 was used in Eqs. (31) and (39) (i.e. 6m␴ instead of 4m␴ ). Chromatographic profiles were reproduced based on the peak shape descriptors taken from the database, using a modified Gaussian function with a parabolic variance (PVMG), which has proved good accuracy [21]. This peak model has the advantage of containing only five parameters that can be easily related to the four peak descriptors. It was proposed as an improvement of the polynomially modified Gaussian model (PMG) [26]. Chromatograms were built by selecting one peak in the database close to the dead time as first peak. The next peak was incorporated by selecting a peak close to: ti+1 = ti + 1.4(Bi,10% + Ai,10% )

(58)

Table 1 Accuracy in peak capacity estimation in the hydro-organic modea Experimentalb

B/Ac

Nc

Eq. (39) GI

Eq. (48) mGI

Eq. (23) mGR

Eq. (31)d mGR

Eq. (31)e mGR

Eq. (31)f mGR

Eq. (56) mI

1.54–25.87 2.15–22.91 2.79–20.29 3.44–18.02

36 32 28 24

1.48 1.41 1.36 1.32

6330 6470 6605 6685

47.8 40.2 33.8 28.4

37.6 32.9 28.4 24.0

40.3 35.0 30.1 25.7

33.7 30.0 26.3 22.8

36.1 32.1 28.1 24.2

35.5 31.6 27.7 23.9

29.0 26.8 24.3 21.6

1.53–10.23 1.84–8.92 2.47–7.75 3.12–6.75

21 18 14 10

1.76 1.75 1.68 1.61

4235 4090 4495 4180

28.1 23.5 17.3 12.0

21.0 17.8 13.6 9.7

23.2 19.7 15.0 10.7

18.8 16.0 12.4 8.9

21.2 18.1 14.0 10.0

20.5 17.4 13.5 9.7

19.8 17.1 13.4 9.7

Time window (min)

Abbreviations: original Giddings’ (GI), modified Giddings’ (mGI), modified Grushka’s (mGR), and modified intuitive approach (mI). a Peak width was measured at 60% peak height for Eq. (39), 50% for Eq. (23), and 10% for the other equations. b Number of peaks that fit in the combined chromatograms, built according to the protocol described in Section 4.2. c Mean values in the selected time window. d Peak variance was obtained according to Eq. (32). e Peak variance was obtained according to Eq. (33). f Peak variance was obtained according to Eq. (34).

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85

Fig. 3. Dependence of left (A, ) and right (B, 䊉) half-widths with retention time for the peaks used to build the combined chromatograms (see Section 4.2), which were eluted with mobile phases of: (a and b) acetonitrile-water, (c) SDS/acetonitrile, and (d) SDS.

Table 2 Accuracy in peak capacity estimation in the micellar modea Eq. (48) mGI

Eq. (23) mGR

Eq. (31)d mGR

Eq. (31)e mGR

Eq. (31)f mGR

Eq. (56) mI

Hybrid micellar mobile phase containing SDS and acetonitrile 1.83–19.43 21 1.28 2660 25.5 2.61–16.05 17 1.25 2770 19.6 3.51–13.28 13 1.23 2790 14.7 4.57–10.94 9 1.21 2845 10.0

21.1 16.9 12.9 8.9

22.2 17.8 13.5 9.3

20.3 16.6 12.8 8.9

21.3 17.3 13.3 9.2

21.1 17.2 13.2 9.2

17.3 14.9 12.0 8.6

Pure micellar mobile phase containing SDS 1.40–12.17 15 1.42 1.70–10.66 13 1.40 2.46–9.30 10 1.38 3.42–7.08 6 1.36

15.1 13.1 10.0 6.0

15.9 13.8 10.4 6.2

13.8 12.1 9.4 5.7

14.7 12.9 10.0 6.0

14.5 12.7 9.9 5.9

12.3 11.2 9.2 5.8

Time window (min)

Experimentalb

B/Ac

Nc

1360 1365 1445 1490

Eq. (39) GI

17.0 14.6 10.8 6.4

Abbreviations: original Giddings’ (GI), modified Giddings’ (mGI), modified Grushka’s (mGR), and modified intuitive approach (mI). a Peak width was measured at 60% peak height for Eq. (39), 50% for Eq. (23), and 10% for the other equations. b Number of peaks that fit in the combined chromatograms, built according to the protocol described in Section 4.2. c Mean values in the selected time window. d Peak variance was obtained according to Eq. (32). e Peak variance was obtained according to Eq. (33). f Peak variance was obtained according to Eq. (34).

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2 Fig. 4. Dependence of total peak variance (tot ) with the squared retention time (tR2 ). See Fig. 3 for details.

This peak was then slightly shifted (less than 10%) up to hold the condition: ti+1 = ti + 1.4Bi,10% + 1.4Ai+1,10%

(59)

which implies that RS = 1.5, since as commented, A and B in the database were measured at 10% peak height: A10% and B10% (i.e. for w = 4.3), instead of 13.4% (i.e. for w = 4). Therefore, z = 1.4 in Eqs. (48) and (56). Three chromatograms were built accordingly, one for hydroorganic mobile phases containing acetonitrile, and two for micellar mobile phases containing SDS/acetonitrile or only SDS. As an example, Fig. 2 shows a chromatogram built for acetonitrile–water mixtures, including the 10 first peaks. 4.3. Estimation of peak capacity for the combined chromatograms A comparative study of peak capacity estimation according to Eqs. (23), (31), (39), (48) and (56) was made based on the chromatograms described in Section 4.2, considering different 2 (needed in Eq. (31)) time windows. The parameters m and ext were obtained by fitting the peak variance (calculated from Eqs. (32)–(34)) against tR2 according to Eq. (30), and the parameters mA ,

mB , A0 and B0 (needed in Eqs. (48) and (56)) were obtained by fitting left and right half-widths versus (tR − t0 ) according to Eqs. (35) and (36). Width at 60% peak height (i.e. 2) was also fitted versus tR (Eq. (6)) to estimate the efficiency assuming ideal peaks (needed in Eq. (39)). The accuracy in the estimations was checked by comparison with the number of peaks that fitted in the combined chromatograms, obtained as explained in Section 4.2 for each time window. The results are shown in Tables 1 and 2 for hydro-organic and micellar mobile phases, respectively. The mean values of asymmetry and efficiency for the peaks in the considered time windows are also given. For the hydro-organic mode, the total time window of the combined chromatogram, which included 36 peaks, was 1.53–25.87 min. Fig. 3a shows the correlations of A and B versus (tR − t0 ) for the time windows 3.81–25.87 min (Fig. 3a) and 1.53–10.23 min (Fig. 3b). As observed, for the narrower time window, the correlations were poorer, especially for B. For the micellar SDS/acetonitrile mobile phases, the total time window was 1.83–19.43 min and included 21 peaks. Finally, for the SDS mobile phases, the total time window was 1.40–12.17 min and included 15 peaks. The corresponding correlations are depicted in Figs. 3c and d, respectively. Except for the right half-width at short retention

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87

Table 3 Peak capacity estimation for several chromatographic systemsa Stationary phase/mobile phase (probe compounds)b

B/Ac

Nc

Eq. (39) GI

Eq. (48) mGI

Eqs. (31)/(33) mGR

Hypersil C18/15% methanol, pH 3 (sulfonamides) Hypersil C18/20% methanol, pH 3 (sulfonamides) Hypersil C18/0.025 M SDS, pH 3 (sulfonamides) Hypersil C18/0.025 M SDS, 6% acetonitrile, pH 3 (sulfonamides) Spherisorb C18/20–60% acetonitrile, pH 3 (celiprolol) Spherisorb C18/17.5–25% acetonitrile, 0.1 M TEA, pH 3 (celiprolol) Spherisorb ODS-2/0.10 M SDS, 4% pentanol, pH 7 (steroids) XTerra C18/15% acetonitrile, pH 3 (␤-blockers) Spherisorb ODS–2/60% acetonitrile, pH 3 (tricyclic antidepressants)

1.65 1.70 1.93 1.35 3.34 2.10 1.56 1.21 5.60

2665 1345 315 1330 165 1625 655 7035 85

17.2 12.6 6.9 11.9 6.1 14.7 8.9 25.3 5.5

16.0 11.4 7.3 12.0 6.9 14.9 8.9 22.6 6.3

16.2 11.6 6.7 11.5 6.0 14.2 8.7 23.2 5.4

See Table 1 for abbreviations. a Peak capacity was estimated in a time window between 2 and 17 min. Peak width was measured at 60% peak height for Eq. (39), and 10% for Eqs. (48) and (31). b Temperature was 25 ◦ C. c Mean values in the selected time window.

times, the plots showed good linearity for A and B versus the retention time, which supports the validity of peak capacity estimation based on Eq. (48). 2 , and t 2 Fig. 4 depicts the correlations between total variance, tot R for the hydro-organic and micellar mobile phases, when the variance was calculated according to Eq. (33). Similar good linearity was achieved according to Eqs. (32) and (34). The results in Tables 1 and 2 demonstrate the good accuracy of Eq. (48) for peak capacity estimation, with prediction errors in the ranges −3.0 to +4.4% and −1.1 to +0.8% for the hydro-organic and micellar modes, respectively. In contrast, the classical approach (Eq. (39)), adapted in this work for RS = 1.5, usually overestimated the number of peaks. Prediction errors with this equation were in the ranges 18.3–33.8% and 6.7–21.4% for the hydro-organic and micellar modes, respectively. We should indicate that the required efficiency values for peak capacity estimation with Eq. (39) were obtained by fitting the widths of several peaks to Eq. (6). It is known, at least in isocratic elution, that the estimations given by Eq. (39) are too peak-dependent. Thus, for instance, in the range 1.54–25.87 for the hydro-organic mode, the estimated peak capacity was 18.0, 47.6 and 49.5 when based on the efficiencies of the peaks at 1.54, 14.0, and 25.9 min. Finding the appropriate peaks to get a representative peak width is rather difficult. Hence, the reported recommendation of averaging the efficiency at diverse retention times, avoiding extreme peaks. To our knowledge, there are at least three reported equations that include the dependence of efficiency with retention in peak capacity estimation (Eqs. (20), (23) and (27)). The integration of Eq. (20) is difficult. Similarly, Eq. (27) is not practical, and the authors recommended the use of a mean plate height (Eq. (28)). Finally, Eq. (23) (adapted here to RS = 1.5) is easy to apply and the results represent an important enhancement with regard to the accuracy

provided by Eq. (39), with errors in the ranges 7–11.9% and 3.3–5.7% for the hydro-organic and micellar modes, respectively. Eq. (23) was derived assuming a linear dependence of peak width at half height (w1/2 ) with retention time [16]. In Eq. (48) or the simplified Eq. (50), a linear dependence was also assumed, but for the half-widths lowered to 10% peak height. From Eq. (47), considering symmetrical peaks (mA = mB = m␴ ): T0 =

w0 w0 = 2m a

(60)

and Pc = 1 +

1 atn + w0 ln za at1 + w0

(61)

which is similar to Eq. (23). Two differences should be commented between Eq. (23) and Eqs. (48) (or (50)): (i) Eqs. (48) and (50) are general equations that can be applied to any value of peak resolution by only selecting the corresponding z value. Eq. (23) was derived assuming exclusively measurement of peak width at half-height. However, at half-height, peak deformation due to asymmetry is less evident than at 10% peak height. This is the reason that explains the larger errors in the measurement of peak capacity yielded by Eq. (23) with regard to Eq. (48). (ii) In Eqs. (48) and (50), ti = tR,i − t0 is used instead of tR,i as in Eq. (23). This shifts the time scale, so that for the dead time, ti = 0, resulting in a positive sign in the linear dependences within the logarithmic term. In this way, the intercept has a clear meaning (w0 or peak width at the dead time). The modified Grushka’s approaches developed for this work (based on Eq. (31), combined with Eqs. (32), (33) or (34)) yielded

Table 4 Peak capacity estimation for several chromatographic systems eluting sets of diureticsa Stationary phase/mobile phaseb ◦

Zorbax SB C18/30% acetonitrile, pH 7, 30 C Zorbax SB C18/30% acetonitrile, pH 3, 60 ◦ C Zorbax SB C18/45% acetonitrile, pH 3, 30 ◦ C Kromasil C18/0.02 M SDS, pH 3 Kromasil C18/0.05 M SDS, pH 3 Kromasil C18/0.15 M SDS, pH 3 Kromasil C18/0.05 M SDS, 10% acetonitrile, pH 3 Kromasil C18/0.05 M SDS, 10% acetonitrile, pH 7 Kromasil C18/0.05 M SDS, 20% acetonitrile, pH 3 Kromasil C18/0.15 M SDS, 10% acetonitrile, pH 3

B/Ac

Nc

Eq. (39) GI

Eq. (48) mGI

Eqs. (31)/(33) mGR

1.07 1.13 1.06 1.55 1.70 1.99 1.33 1.39 1.27 1.15

6695 7584 7246 1486 1077 561 3632 3390 4980 3020

24.0 25.9 24.9 12.9 11.4 8.9 18.9 18.5 21.7 16.7

21.4 23.2 22.3 12.2 11.4 9.5 17.5 16.8 18.4 14.8

22.1 23.9 23.0 12.2 11.0 8.7 17.7 17.2 19.1 15.3

See Table 1 for abbreviations. a Peak capacity was estimated in a time window between 2 and 17 min. Peak width was measured at 60% peak height for Eq. (39), and 10% for Eqs. (48) and (31). b Temperature was 25 ◦ C, unless indicated. c Mean values for the peaks in the considered time window.

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prediction errors in the ranges −5.0 to −11.4%, 0 to +1.0%, and −0.4 to −3.6%, respectively, for the hydro-organic mode, and −1.1 to −8.0%, −2.0 to +2.3%, and +0.5 to +3.3%, respectively, for the micellar mode (Tables 1 and 2). As noted, the best results were obtained when the total variance was measured according to Eq. (33), improving slightly the results obtained with the modified Giddings’ approach (Eq. (48)). Finally, the simplified approach based on mean peak widths (Eq. (56)) underestimated the number of peaks with errors in the ranges 3–19.4% and 3.3–17.6% for the hydro-organic and micellar modes, respectively (Tables 1 and 2). The reason for the underestimation is the larger weight given to wider peaks. 4.4. Peak capacity for real chromatograms Peak capacity was estimated for several chromatographic systems (i.e. combination of column and mobile phase), using the peak profile information obtained for several probe compounds. Tables 3 and 4 show the results obtained with Eq. (39) and (48), and for Eq. (31) combined with Eq. (33), for hydro-organic and micellar systems in a time window (tR − t0 ) = 2 to 17 min. We decided taking an initial peak at 2 min, because the behaviour of the right half-width (B) was non-linear at times closer to t0 (1.2 min), as shown in Figs. 1 and 3. Also, measurement of peak half-widths in the neighbourhood of t0 at 10% peak height is rather inaccurate. 5. Conclusions Peak capacity is a useful concept, but it can be only accessible through calculation, since it is based on a situation rarely met in practice: a chromatogram with well distributed peaks and resolved without gaps. Proper peak capacity estimation requires knowledge on how peak width changes with retention time. Several equations have been proposed to make these estimations, which were outlined under three different perspectives, represented by Eq. (1) (intuitive approach), Eq. (7) (Giddings’) and Eq. (18) (Grushka’s). The intuitive approach assumes the same width for all peaks in the chromatogram, whereas in the equations originally proposed by Giddings and Grushka, the efficiency was kept unchanged with retention time. This assumption may not, however, yield correct estimations. Real chromatograms that follow the definition of peak capacity are not easy to find. Thus, we checked the estimations on combined chromatograms built with real peaks. The classical equation for peak capacity estimation (Eq. (39)) tended to yield overestimations, with minor errors for time windows including a smaller number of peaks. Thus, interestingly, in the MLC examples where the efficiencies were smaller, the errors found were smaller. A proper inclusion of changes in peak width with retention time is not straightforward for either modified Giddings’ and Grushka’s approaches. On the one hand, the equation relating the total peak 2 , with t 2 is not compatible with Giddings’, and approxivariance, tot R mate linear equations between each half-width and retention time should be used. The magnitude of the curvature of the parabola depicted by  tot versus tR seems to be, however, rather small and can be assimilated to a straight-line (see Figs. 1 and 3). This approximated linear dependence has proved to be useful in the prediction of peak shape, which is of interest in the optimization of chromatographic selectivity [21]. On the other hand, the estimation of the standard deviation for an asymmetrical peak is not straightforward, and different approximations should be used. We have applied three alternatives in this work (Eqs. (32)–(34)). In spite of the commented limitations, both approaches Eq. (31) combined

with Eq. (33), and Eq. (48) seem to yield rather accurate estimations of peak capacity. For both approaches, measurement of peak width was made at 10% peak height to account peak deformation better. It should be noted that the final equation derived from the modified Giddings’ approach coincides with that obtained with a previous modified Grushka’s approach (Eq. (23)), where the linear approximation between standard deviation and retention time is assumed, with the restriction that peak widths should be measured at 10% peak height. The equations previously proposed for peak capacity estimation, which incorporate the dependence of efficiency with retention time (as the examples given in this work: Eqs. (20), (23) and (28)), were outlined based on the Grushka’s approach. These equations, together with Eq. (31) developed for this work, do not allow including peak skewness as the Giddings’ approach (Eq. (48)) does. It should be observed, however, that in the simplified equation (Eq. (50)) the difference between left and right half-widths is not manifest anymore. This also holds for Eq. (56). This, together with the low errors provided by Eq. (50), leads to the conclusion that peak width (and not skewness) is the relevant factor, at least in the studied examples. Nevertheless, in other situations (such as for highly distorted electrophoretical peaks), asymmetry could be a key factor. Nomenclature

A and B left and right half-widths A0 and B0 left and right half-widths at the dead time H plate height k retention factor L column length m slope of the relationship of standard deviation versus time mA and mB slope of the relationships between left and right halfwidths versus time n number of peaks N efficiency Pc peak capacity sample peak capacity Pc∗∗ RPLC reversed-phase liquid chromatography classical resolution RS SDS sodium dodecyl sulfate t time window t time t0 dead time tD dwell time tG gradient run time tR retention time w peak width peak width at the dead time w0 w1/2 peak width at half-height z number of standard deviations  peak standard deviation 2 col column variance 2 ext extra-column variance 2 tot observed peak variance ϕ organic solvent content (%, v/v) Acknowledgements This work was supported by Projects CTQ2004–02760/BQU and CTQ2007–61828/BQU (Ministerio de Educación y Ciencia of Spain, MEC) and FEDER funds. SPT thanks a FPI grant from the MEC. The authors thank one of the reviewers for several interesting

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