Journal of Chromatography A, 1386 (2015) 81–88

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

Kinetic plots for gas chromatography: Theory and experimental verification Sander Jespers a , Kevin Roeleveld b , Frederic Lynen b , Ken Broeckhoven a , Gert Desmet a,∗ a b

Vrije Universiteit Brussel, Department of Chemical Engineering, Pleinlaan 2, 1050 Brussels, Belgium Universiteit Gent, Separation Science Group, Krijgslaan 281, B-9000 Gent, Belgium

a r t i c l e

i n f o

Article history: Received 19 December 2014 Received in revised form 16 January 2015 Accepted 16 January 2015 Available online 23 January 2015 Keywords: Gas chromatography Kinetic performance limit Optimal pressure Efficiency Thin-film

a b s t r a c t Mathematical kinetic plot expressions have been established for the correct extrapolation of the kinetic performance measured in a thin-film capillary GC column with fixed length into the performance that can be expected in a longer column used at the same outlet velocity but at either the maximal inlet pressure or at the optimal inlet pressure, i.e., the one leading to an operation at the kinetic performance limit of the given capillary size. To determine this optimal pressure, analytical solutions have been established for the three roots of the corresponding cubic equation. Experimental confirmation of the kinetic plot extrapolations in GC has been obtained measuring the efficiency of a simple test mixture on 30, 60, 90 and 120 m long (coupled) columns. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In a series of papers [1–3], two different methods have been established to directly calculate and compare the kinetic performance limit (KPL) of different liquid chromatography (LC) systems, in short referred to as the kinetic plot method. A kinetic performance limit plot directly relates the optimized separation efficiency to the minimal required analysis time. This information obviously is more of a practical relevance, while the traditional Van Deemterplot (plate height versus mobile phase velocity) is more suited for theoretical analysis. More specifically, the kinetic plot method allows to recalculate a set of experimental measurements of time and efficiency obtained on a column with fixed length into a set of data points forming the KPL of the chromatographic support in the test column. This KPL combines the band broadening as well as the pressure drop characteristics of the support into a single curve, thus providing a unique signature of its kinetic performance under a given set of operating conditions (mobile phase composition, column type, temperature). The single assumption underlying the method is that the plate height is independent of the column length, an assumption which in LC is theoretically sound, at least when neglecting some practical length-dependent issues. A detailed discussion of these issues can be found in literature [3–7].

∗ Corresponding author. Tel.: +32 26293251; fax: +32 26293248. E-mail address: [email protected] (G. Desmet). http://dx.doi.org/10.1016/j.chroma.2015.01.053 0021-9673/© 2015 Elsevier B.V. All rights reserved.

Whereas earlier work on kinetic plots [8–10] used iterative algorithms to construct plots of time versus efficiency, and was therefore limited to theoretical data, the simple kinetic plot expressions developed in [1–3] for liquid chromatography only involve a few multiplications and divisions and can be readily applied to any set of experimental data. As a consequence, the method has now become quite popular and has been used by many authors to assess the true kinetic merits of the systems they are developing or investigating [11–33]. The first of the two methods proposed in [1–3], further referred to as the plate height-based method, rely on the measurement of the plate height Hobs , the velocity uo of the non-retained component marker and the column permeability. The second method, referred to as the column elongation-method, directly uses the tm -time of the non-retained component and the corresponding separation efficiency N or peak capacity (nP ) measured on a given column. Whereas the first method is better suited to understand and visualize the link between the traditional Van Deemter curve and the KPL-curve, the advantage of the second method is that it also works for programmed temporal changes in the mobile phase composition (gradient LC) as rigorously shown in [3] and applied in [6,18,22,24,26–29,31,34]. The column-elongation method also works for conditions where the mobile phase velocity undergoes temporal changes, as occurs for example in constant-pressure gradient LC [35,36]. Whereas the work in [1–3] was restricted to LC, where the spatial pressure-gradient to a first approximation scales linearly with

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the column length, the question remained to which extent the established methods can still be used in cases with a significant mobile phase compressibility, as occurs in SFC and GC. In SFC, the additional strong dependency of retention on pressure (through the mobile phase density) obviously complicates the extrapolation and prohibits the derivation of an analytical mathematical expression. However, by measuring the performance as a function of flow rate for a fixed average column pressure (i.e., the so-called isopycnic method), a good approximation of the kinetic performance limits could be obtained [25,30,37]. Seminal work on the kinetic optimization of GC separations has been carried out by, amongst others, Giddings [8,38], Cramers [39–41] and Blumberg [42,43]. Kurganov et al. [44] recently proposed to extend the plate-height based kinetic plot method to account for the compressibility effects occurring in the case of ideal gas chromatography (GC). They applied their method to describe the kinetic performance of a set of monolithic GC columns used under isothermal conditions [44]. In the present study, we reviewed their work and also extended the column elongation-variant of the kinetic plot method to GC. This lead to expressions for the extrapolation to the maximal inlet pressure (Section 3.1), as well as for the extrapolation to the optimal inlet pressure (Section 3.3). In LC, both extrapolations are the same. In GC, they differ because of the inlet pressure dependency of the observed plate height when different column lengths operating at the same outlet velocity are compared. 2. Experimental All chemicals were HPLC grade from Sigma–Aldrich (St. Louis, MO, United States). 4 HP-5 MS columns (30 m × 250 ␮m × 0.25 ␮m) were obtained from Agilent (Santa Clara, CA, United States). An Agilent 6890 gas chromatograph with FID detector and split/split less injection was used. The H2 carrier gas was supplied by a Parker Balston Hydrogen Generator H2PD-300-220 (Haverhill, MA, United States). The maximum pressure drop of this system was 184 kPa due to limitations on the gas generation rate. Polyimide sealing resin from Grace Davison Discovery Sciences (Columbia, MD, United States) and universal 2-way fused silica unions from Agilent were used to couple the columns according to the included instructions. The test mixture contained ethyl-caprate, tridecane and pentadecane dissolved in 2,2,4-trimethylpentane at a concentration of 50 ppm for each component. A headspace sample was made to determine the elution time of 2,2,4-trimethylpentane, while a separate sample of 50 ppm was made for each of the three components to determine their elution order. Injection of 1 ␮L sample was done at 250 ◦ C and 20:1 split ratio. Separations were performed under isothermal conditions with the flow varying between 0.2 and 1.2 mL/min and oven temperature set at 100 ◦ C. The detector temperature was set at 300 ◦ C, H2 flow at 40 mL/min, air flow at 300 mL/min and makeup flow at 20 mL/min. Data was analyzed with HPCore ChemStation. Measurements on the 120 m column were performed using a mixture containing 100 ppm of each component and a split ratio of 10:1 to increase sensitivity. 3. Derivation of the theoretical expressions 3.1. Expressions for the extrapolation to maximal inlet pressure performance 3.1.1. Plate height-based extrapolation method As shown in [1,2], the performance on a column with given length Lexp and operated with a pressure difference pexp can be directly extrapolated to the performance on a column operating at

the maximal pressure drop but with the same outlet velocity by combining the three basic equations for efficiency, mobile phase velocity and pressure drop in chromatography, adapted here to account for the specific compressibility of the carrier medium in ideal gas GC [45] (see Supplementary Material SM for details): Lexp Hobs

Nexp =

(1)

3 − 1) Lexp Lexp 2(Pexp = 2 − 1) uo,exp 3(Pexp ut,exp

tm,exp =

pexp = =

(2)

Lexp ux,exp Lexp uo,exp 2(Pexp − 1) = 2 − 1) Kv Kv (Pexp Lexp uo,exp f0,exp Kv

(3)

wherein Hobs is the observed plate height, L the column length, Nexp the observed plate number, tm,exp the experimentally observed elution time of a non-retained compound (in LC this is often written as t0 ), pexp the experimental pressure drop over the column, uo,exp the mobile phase outlet velocity,  the mobile phase dynamic viscosity, Kv the column permeability and Pexp the ratio of inlet pressure to outlet pressure during the experiment: Pexp =

pi,exp po,exp

=

pexp + po,exp po,exp

(3b)

Because of the mobile phase compressibility, the carrier gas in GC expands from the high pressure inlet toward the low pressure outlet. One of the consequences of this expansion is that the elution time of the non-retained compound (tm,exp ) is determined by u¯ t , the time-averaged mobile phase velocity (Eq. (S-4) in SM), whereas the pressure drop (pexp ) depends on u¯ x , the space-averaged mobile phase velocity (Eq. (S-11) in SM) [39,40,46]. In an incompressible liquid, both velocities are equal, such that this distinction does not need to be made in LC (even under UHPLC conditions where the liquid displays some compressibility, the effect is minor). For a fluid following the ideal gas law, the relation between u¯ t and u¯ x is given by Eqs. (S-12) and (S-13) of the SM, and can also be found in literature [47]. In GC, the observed plate height can generally be written as [42,48,49]. Hobs = H0 f1,exp + Cs uo,exp f2,exp

(4)

with: Ho =

B + Cm uo,exp po,exp po,exp uo,exp

f1,exp =

f2,exp =

4 − 1)(P 2 − 1) 9(Pexp exp 3 − 1) 8(Pexp 2 − 1) 3(Pexp 3 − 1) 2(Pexp

2

(5)

(6)

(7)

Extrapolating an experimentally measured efficiency on a column with given length into an extrapolated efficiency expected in a column containing the same support but now elongated such that it generates a given maximal pressure drop while keeping the same outlet velocity is only correct if this extrapolation leaves the plate height Hobs unchanged. This is (approximately) true in LC, but clearly not in GC, given the inlet pressure dependency of Hobs via f1 and f2 in Eq. (4). However, making the approximation that the film mass transfer is negligible (Cs very small), which holds for most open-tubular GC systems, the dependency which exists between Hobs and column length at the same outlet velocity uo can be expressed using a single flow-dependent factor H0 and a single

S. Jespers et al. / J. Chromatogr. A 1386 (2015) 81–88

pressure-dependent factor f1 accounting for the gas compressibility. As shown in the SM, the explicit expressions for the plate heightbased extrapolation to the maximal operating pressure for a thinfilm GC column are given by:

 Npmax =

 =



pmax Kv ·  uo H0

tm,pmax =



pmax ·  pmax 





exp



2

4 9(Pmax − 1)(Pmax − 1)

· f3,max





·

u2o

exp



u2o

3 4(Pmax − 1)

(8)

exp

Kv

Kv

·



pmax Kv .  uo H0



=







3 (Pmax − 1) 3(Pmax − 1)

· f4,max

(9)

exp

(N, tm , L) has to be scaled with a different elongation factors because each has a different dependency on the pressure. The column elongation-method is based on the direct physical interpretation of the pmax -extrapolation process, transforming a given efficiency (Nexp ) obtained in a given time tm,exp on a column with length Lexp and producing a given pressure drop pexp into the performance one can expect in a column with a maximized length Lpmax , defined as the column length that would produce the maximally allowable pressure drop pmax while keeping the same mobile phase outlet velocity. Defining the corresponding column length elongation via the elongation factor 1 (see Eq. (S-17)), the corresponding increase in tm and N can subsequently be calculated using two other elongation factors (2 and 3 ), as shown below: Lpmax = 1 Lexp

(11)

tm,pmax = 2 tm,exp

(12)

Npmax = 3 Nexp

(13)

As shown in the SM, the expressions for the pressure dependency of the  – elongation factors for thin-film GC are given by:

2

4(P 3 − 1) (P 3 − 1) with f3 = and f4 = . 3(P − 1) 9(P 4 − 1)(P − 1) The expressions between the round brackets in Eqs. (8) and (9) represent the part of the solution already obtained for the incompressible liquid case (kinetic plot expressions for LC [1]), whereas the part outside the brackets accounts for the gas compressibility. The difference between Eqs. (8) and (9) and those presented by Kurganov et al. [44] is that Eqs. (8) and (9) are written as a function of the outlet velocity, hence allowing to properly account for the pressure dependencies of u¯ t , u¯ x and Hobs via the factors f3 and f4 . To use Eqs. (8) and (9), one should first derive the values for uo and H0 from the experimentally measured u¯ t and Hobs using respectively Eqs. (S-12) and (S-5a) of the SM. Given the very low dependency of the retention factor k on the pressure in GC (similar to the LC-case), k can be expected to remain unchanged between the experimental and the extrapolated conditions. The analyte retention time can hence be readily derived from the tm -time (cf. Eq. (9)) using: tR,pmax = tm,pmax (1 + k)

83

(10)

The error made by making the thin film-approximation, i.e., neglecting the second term in Eq. (4), is very small, even when Cs is significant. This is due to the fact that the pressure dependency of f1 and f2 is opposite, such that even when the film-thickness is in the order of 1% of the capillary radius (±100 times larger than the filmthickness in commercially available capillary thin-film GC columns and outside of the scope of the definition of thin-films proposed by Blumberg [50]) the error on Hobs is only of the order of 8% (similar results were obtained by [51]). For systems with a significant Cs -contribution (such as packed bed columns or thick film capillaries), a direct extrapolation of a given experimental Hobs -data point is no longer possible because the length-dependency of Hobs depends upon two factors (f1 and f2 ) with a variable weight (determined by the relative value of H0 and Cs ). In this case, one has to resort to the fitting method described in Section 4 (Eq. (28)), where the constants B, Cm and Cs are first determined by fitting the complete experimental van Deemter curve (which could also include an A-term). 3.1.2. Column elongation-based extrapolation method The column elongation-based kinetic plot method developed in [3] for LC uses a single column elongation factor to directly calculate the kinetic performance limit of a given chromatographic system. In GC, however, each of the different performance characteristics

1 =

2 =

2 pmax ((Pmax − 1)/(Pmax − 1)) 2 − 1)/(P pexp ((Pexp exp − 1)) 3 pmax ((Pmax − 1)/(Pmax − 1)) 3 pexp ((Pexp

− 1)/(Pexp − 1))



=

 =

pmax pexp pmax pexp

 

f0,exp f0,max

(14)

f4,max f4,exp

(15)

2

3 =

3 4 pmax ((Pmax − 1) /((Pmax − 1)(Pmax − 1))) 2

 =

3 − 1) /((P 4 − 1)(P pexp ((Pexp exp − 1))) exp

pmax pexp



f3,max f3,exp

(16)

wherein f0,max , f4,max , f3,max and f0,exp , f4,exp , f3,exp (Eqs. (3) and (9)) are correction factors corresponding to the maximum and experimental pressure drop, respectively. Eqs. (11)–(13) can be readily applied to any set of N and tm -data measured on a column with given length Lexp . 3.2. Determination of the optimal inlet pressure In LC, the necessary and sufficient condition for a given chromatographic support to operate at its kinetic performance limit (KPL) is that it is operated at the maximal pressure drop [1,3]. In GC, this is no longer true. In the SM, it is shown how maximizing N for given tm or minimizing tm for a given N (which are the two equivalent criteria defining the kinetic performance limit) for the case of thin-film GC corresponds to: N = Nmax for a fixed tm ⇔

∂[Hobs

= 0



3 − 1)/(3p 2 (4(Pexp exp (Pexp − 1)(Pexp + 1)))]

∂pexp (17)

Fig. 1a shows that the function between straight brackets goes through a minimum at a finite value of pexp . Approximating the problem by leaving out the f1 -dependency of Hobs (f1 is close to unity for most applications under atmospheric outlet and varies over maximally 12.5%, see SM for a more precise error estimation), searching the minimum of the function between straight brackets corresponds to solving the following cubic equation (see SM):

84

(a)

S. Jespers et al. / J. Chromatogr. A 1386 (2015) 81–88

2.0

This expression is equivalent to the result obtained by [41,43], who found that the kinetically √ optimized outlet velocity for a system operating at high P is 2 times larger than the outlet velocity corresponding to the minimum of the Hobs curve. When tm is sufficiently large, the optimal pressure drop defined by either Eqs. (19) or (20) may exceed the maximal system pressure. In that case, the (practically achievable) kinetic optimum coincides again with the maximal inlet pressure performance (popt = pmax ).

1.5 1.0 0.5 0.0

0

50

100

150

200

∆pexp (kPa)

3.3. Expressions for the extrapolation to optimal inlet pressure performance

(b) 400 300 200 100 0

200

0

Fig. 1. (a) Plot of Hobs

400



3 4(Pexp

−

600

800

tm (s)

1000

1200



2 1)/((3pexp (Pexp

− 1)(Pexp + 1)) versus pexp for

a fixed value of tm = 15 s. (b) Real part of the solutions to Eq. (18). Full line: popt,1 ; dotted line: popt,2 . At the discontinuity in the functions around (tm = 60 s) popt,1 changes from a complex to a real number and the opposite occurs for popt,2 . Calculations were based on the B- and C-values obtained by fitting the van Deemtercurve for ethyl-caprate on a 30 m column (Kv = 1.95·10−9 m2 ,  = 1.34·10−5 kg/ms, po = 101,325 Pa).

p3exp + ˛p2exp − ˇ = 0 with ˛ = 3po,exp , ˇ =

 = Cm po,exp

(18)

6Bp2o,exp Cm

, B =

The expressions from Section 3.2 allow to calculate the optimal pressure drops popt for any set of freely chosen tm -times. This set can unfortunately not be directly linked to the corresponding uo velocities (a numerical iteration would be needed to do so). This implies that the procedure for the direct calculation of the KPL is limited to establishing fitted KPL-curves (because tm can be treated as a freely selectable variable in this case) and not to a set of experimental data points. To establish this best-fit curve, one can start from the fitted B- and Cm -values of the measured H0 -curve and replace pmax by popt in Eq. (8):

B po,exp



NKPL =

popt Kv uo H0,exp

 =

popt,1

3 =

Kv tm

⎡  √

3 − 3 3 27ˇ2 − 4ˇ˛3 − 27ˇ + 2˛3 ⎣ −  √ 3 2

3

√ 3 √

(1 + i 3) 3 3 27ˇ2 − 4ˇ˛3 − 27ˇ + 2˛3 = + √ 3 6 2



popt pexp

3

6Bp2o,exp Cm

 · f3,opt

(23)



(24)

f3,opt f3,exp

(25)

⎤

√ 3 2˛2

√

3 3 27ˇ2 − 4ˇ˛3 + 2˛3



− ˛⎦

√ (1 + i 3)˛2

√ √

3 3 3 4 3 3 27ˇ2 − 4ˇ˛3 − 27ˇ + 2˛3



√ (1 − i 3)˛2

√ √

3 3 3 4 3 3 27ˇ2 − 4ˇ˛3 − 27ˇ + 2˛3

Note that popt is a function of tm via the factor ˇ. For small tm , only the popt,1 -root is a real number and should hence be used. For large tm , the popt,2 -root becomes the single real root (see Fig. 1b). The root popt,3 relates to a physically impossible solution involving a decreasing optimal pressure drop for an increasing tm . When P approaches infinity, ˛ = 0 in Eq. (18) and the three roots converge to a single one, given by: popt =

4 − 1)(P 9(Popt opt − 1)

with popt given by Eqs. (19) and (20).

√ 3 √

(1 − i 3) 3 3 27ˇ2 − 4ˇ˛3 − 27ˇ + 2˛3 = + √ 3 6 2



2

Similarly, one can also start from the fitted expression of Nexp and use the 3 -factor from Eq. (16) but with popt instead of pmax . Fitting Nexp is done using Eq. (28) (see further on).

with



popt,3

3 − 1) 4(Popt

NKPL = 3 · Nexp

tm and Cm Kv



popt,2

·

popt Kv uo ((Bexp /(po,exp uo )) + (Cm,exp po,exp uo ))

Calculating the roots of this cubic equation using the procedure described in [52], we obtain one real and two complex roots: 1 = 3



(22)

(19)

−

˛ 3

(20)

−

˛ 3

(21)

4. Results and discussion 4.1. Efficiency measurements Fig. 2a–c shows the chromatograms obtained on respectively 1, 2 and 4 coupled columns at maximum pressure. The isothermal efficiency of the columns measured at 100 ◦ C respectively corresponded to 49,700, 175,500, and 480,500 for the second component

S. Jespers et al. / J. Chromatogr. A 1386 (2015) 81–88

(ethyl-caprate). Similar values were obtained for the other components.

and



tm,pmax =

pmax 



4.2. Maximal pressure extrapolations Fig. 3c compares the pmax -extrapolations obtained via the plate height-based method (i.e., starting from the Van Deemterplot in Fig. 3a) with that obtained via the column elongation-based kinetic plot method (i.e., starting from the fixed length-plot of tm versus Nexp represented in Fig. 3b). First considering the data points, it can be seen that the plate height- and the column elongation-based method lead to the same extrapolated values (with a small exception for the points at the highest N), showing that both methods are fully equivalent. In fact, the equivalence between the plate-height based method (Eqs. (8) and (9)) and the column elongation method (Eqs. (12) and (13)) can be mathematically established by starting from Eqs. (8) and (9) and substitute pmax f3,max and pmax f4,max by 3 pexp f3,exp and 2 pexp f4,exp respectively (i.e., using Eqs. (15) and (16)). As can be noted, this transforms Eqs. (8) and (9) directly into Eqs. (12) and (13).

 Npmax =



pmax Kv  uo H0



= 3





· f3,max exp



pexp Kv  uo H0

(a)





· f3,exp = 3 · Nexp

20

16

= 2

u2o

pexp 





· f4,max exp



Kv



u2o

 · f4,exp = 2 · tm,exp

(27)

exp

8 7

12

6

10

AU (/)

8

N= 49700

N= 43800

N= 176000

5 4 3

4

2

2

1

0

(c)

Kv

tm= 2.24

9

14

6



The small deviation in the high N-end is due to the different data sets upon which the models are based. Whereas the column elongation method requires only pressure data, the plate height method requires both pressure and velocity data. The errors on the velocity, although small, lead to the observed difference at the high N-end, where these errors have a higher weight than at the low N-end. The curves fitted through the data points can be established using the same equations. Again two approaches can be followed: one via the plate-height method and one via the elongation factormethod. In the former case, it is obvious to first fit a set of plate height data such as the one shown in Fig. 3a using the theoretical expression for the plate height in open-tubular isocratic GC (Eq. (5)) to obtain the best fit values for the B- and Cm -parameters. Subsequently using these values to generate a set of fitted (Hobs ,uo )-data and inserting these into Eqs. (8) and (9) yields a best-fit curve for the pmax -extrapolated data points (cf. the full line added to Fig. 3c). Using the column elongation-method on the other hand, it is more convenient to start from a set of measured Nexp and tm data (cf. Fig. 3b) and fit these using the following theoretical relationship

(b) 10

tm= 0.60 N= 53300

18

AU (/)

(26)

exp

85

N= 175500

N= 156400

0 0

10

5

t (min)

10

15

0

20

t (min)

40

60

tm= 9.04

9

N= 476700

8 7 6

AU (/)

5 4

N= 480500

3

N= 452300

2 1 0 0

100

200

t (min) Fig. 2. Chromatograms of the separation of tridecane (k = 7.1), ethyl-caprate (k = 13.4) and pentadecane (k = 25.8) in 2,2,4-trimethylpentane at 100 ◦ C and p = 184 kPa on (a) 30 m, (b) 60 m, and (c) 120 m columns. Measurements on the 120 m long column were performed using a 2 times higher sample concentration and split ratio compared to the other columns.

86

S. Jespers et al. / J. Chromatogr. A 1386 (2015) 81–88

Fig. 3. (a) Van Deemter-curve for tridecane obtained on a 30 m column, plotting Hobs versus uo . The experimental date points were fitted using Eqs. (4) and (5), with Cs = 0. (b) Fixed length kinetic plot for tridecane plotting tm versus N on a 30 m column, data points were fitted using Eq. (28). (c) pmax -extrapolated variable length kinetic plots obtained with the plate height-based extrapolation method (open symbols and dashed line) and the column elongation-method (full symbols and line); both methods were applied to the experimental data as well as the fitted curves. Experimental measurements on the 60 m, 90 m and 120 m column at maximum pressure drop are represented by the asterisks.

Table 1 Comparison of values for the constants appearing in the Van Deemter curve (Eq. (4)) and the fixed length kinetic plot (Eq. (28)). Van Deemter curve (Eq. (4))

B (m·kg/s3 ) Cm (m·s3 /kg)

Fixed length kinetic plot (Eq. (28))

Tridecane

Pentadecane

Ethylcaprate

Tridecane

Pentadecane

Ethylcaprate

4.52 2.98·10−9

3.92 3.57·10−9

4.32 3.10·10−9

4.60 3.08·10−9

3.82 3.56·10−9

4.36 3.12·10−9

which can be established between time and efficiency: Nexp =

Lexp tm 2 )/(p L ((Bf1,exp f2,exp tm o exp )) + ((Cm f1,exp po Lexp )/(f2,exp ))

(fixed length kinetic plot)

(28)

Eq. (28) can be derived by combining Eqs. (1) and (4) with the relation between the outlet and the time-averaged mobile phase velocity (Eq. (S-12) of the SM). Fitting Eq. (28) to the experimental Nexp and tm -values shown in Fig. 3b yields a value for the B- and Cm -constants appearing in it. Obviously, if there would be no measurement or modeling errors, these B- and Cm -values should be equal to those obtained via the plate height fit (i.e., via Eq. (5)). As shown in Table 1 comparing the B- and Cm -constants for the different analytes considered in the present study, this indeed holds to a large degree. The small deviations are caused by the experimental errors and the different weights these errors inevitably have in the numerical fitting procedures used to fit the Hobs ,uo -data versus the fit of the tm ,Nexp -data set. When the fitted B-values are used to calculate Dmol (according to the expression for the B term in the Golay equation [48]) and when these Dmol -values are subsequently used together with the fitted Cm -values to calculate the capillary radius rcap (using the expression for the Cm -term in the well-established Golay equation) an average of 129 ␮m ± 5 ␮m is obtained which is in good agreement with the expected 125 ␮m. With B and Cm known (from either the fit of the Hobs ,uo -data or from the tm , Nexp -data set), one can also directly draw a best fit curve for the pmax -extrapolated data in Fig. 3c (dashed line). Introducing the parameter amax , defined as amax = pmax Kv / to shorten the notation, this can be done by inserting these B- and Cm -values into the following expression describing the relation between N and tm in a column operating at maximal pressure: Npmax =

2a tm max (f2,max /f0,max ) 2 ((Bf1,max f2,max tm )/po ) + ((Cm f1,max po amax tm )/f0,max )

(29)

Fig. 4. Zoomed view of the overlay of the fixed length kinetic plot on the 30 m column (full symbols and full line) and the pmax -extrapolated kinetic plot (open symbols and dashed line) calculated using the plate height method for tridecane. Experimental measurements on the 60 m, 90 m and 120 m column at maximum pressure drop are represented by the asterisks.

Eq. (29) is obtained by combining Eqs. (1)–(5) and replacing the velocity appearing in the expression for H0 by the time-averaged velocity (Eq. (S-12)) and Lexp by its dependence on column pressure drop and tm (Eq. (S-14b)). While the good agreement between the plate height-based and the column elongation-based methods proves their mutual consistency, the fact that the experimental measurements at pmax for the 60 m, 90 m and 120 m columns (see * data points) coincide exactly with the predicted extrapolation curves is even more important, as it provides a direct experimental proof for the pmax extrapolation expressions given by Eqs. (8), (9), (15) and (16). 4.3. Optimal pressure extrapolations (kinetic performance limit (KPL) curves) The fact that the fixed length kinetic plot (solid line) descends below the pmax -extrapolation (dashed line) in Fig. 4 readily shows that working at the kinetic optimum (=reaching a given N in the shortest time) in GC is not equivalent to working at the maximum pressure drop, as is the case in LC. Noting that the kinetic performance limit (KPL) is in GC obtained at some finite inlet pressure is of course in agreement with the popt -calculations in Section 3.2. Following the procedure

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Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.chroma. 2015.01.053.

References Fig. 5. Overlay of the fixed length kinetic plot (dashed line), the pmax -extrapolated kinetic plot (dotted line) and the kinetic performance limit (full line) calculated using the plate height-based method (Eq. (23)) for ethyl-caprate at popt . Experimental data points for the 30 m column (full symbols) are shown. The asterisk represents the measurements at popt (100 kPa for 30 m).

described in Section 3.2, the full line curve shown in Fig. 5 is obtained. Fig. 5 provides a zoom-in of Fig. 4, and clearly shows that the full line curve obtained via the expressions established in Sections 3.2 and 3.3 indeed corresponds to the curve enveloping all possible fixed-length curves, whereas the pmax -extrapolated curve clearly cuts through the fixed-length performance of the 30 m column. At large tm (i.e., for separations conducted in longer columns) the difference between the popt and the pmax extrapolation vanishes (cf. the merge between the full and the dashed line). 5. Conclusions Unlike LC, the pressure-dependency of the different terms contributing to Hobs in GC prohibit the establishment of generally valid kinetic plot expressions extrapolating measurements on one column length into the performance in a column with extrapolated length and operated at the maximal or the optimal inlet pressure. However, for the special case of thin-film capillary columns, the pressure dependency is expressed via a single analytical expression. In this case, the extrapolation expressions can still be written in an analytical form using either the plate height-based method or the column elongation method. Another feature distinguishing GC from LC is that the kinetic performance limit of a column format with given size is not obtained when applying the maximal inlet pressure, but at some smaller, optimal pressure (provided this optimal pressure does not exceed the maximally allowable pressure of the system). This optimal pressure can be found as one of the real roots of a cubic equation and depends on tm , as well as on some column- and component properties. For small tm times the optimal pressure drop is lower than the maximum pressure drop, whereas for high tm times the optimal pressure drop surpasses the maximum pressure drop so that the latter has to be used (as is the case for LC). Checks of theory have been made by comparing pmax extrapolations based on the plate height method and the column elongation method, as well as by comparing the pmax extrapolation with the KPL. Very good agreement between the plate height-based method and the column elongation method are observed, with only small differences appearing at very large tm times. The difference between the pmax -extrapolation and the KPL was found to be the largest as small tm times, and faded out completely at tm = 160 s and above. The pmax -extrapolation and KPL, constructed based on the performance measured on one 30 m column, were verified experimentally on coupled columns. Acknowledgement S.J. gratefully acknowledges Research grant from the Research Foundation Flanders (FWO Vlaanderen).

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