A calculation method for the prediction of effective plate height in capillary gas chromatography

Journal of Chromatography A, 1216 (2009) 8986–8991

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A calculation method for the prediction of effective plate height in capillary gas chromatography P. Moretti ∗ , S. Vezzani, G. Castello University of Genoa, Dipartimento di Chimica e Chimica Industriale, Via Dodecaneso 31, Genova I-16146, Italy

a r t i c l e

i n f o

Article history: Received 17 June 2009 Received in revised form 22 October 2009 Accepted 23 October 2009 Available online 31 October 2009 Keywords: Capillary columns Non-polar phases Gas chromatography Isothermal analysis Mathematical models Effective plate height Plate height Retention time Peak width at half height

a b s t r a c t The effective plate height, heff , is considered to be a better measure of the efficiency of capillary column than the conventional plate height, h, in isothermal conditions. By using experimental data of 1-alcohols and n-alkanes, 2-ketones and 1-alkenes measured on capillary columns coated with non-polar stationary phases in isothermal and isobaric conditions, the peak width at half height is predicted with a function similar at that of adjusted retention time. The results obtained under different analytical conditions as the head pressure and the temperature of the column confirm the validity of the model, whose parameters are linear, and as a consequence a unique solution is obtained. © 2009 Elsevier B.V. All rights reserved.

1. Introduction In previously published papers [1–3], a method was described which evaluates the column efficiency and predicts the retention times in gas chromatographic linear gradient programmed pressure isothermal runs by using the retention times and the widths at half height of peaks obtained in few isobaric runs, by taking into account the diffusion coefficients of the analysed compounds into the mobile and stationary phase. In a further work [4], the same method was used for the automatic prediction of the plate number in isothermal analyses carried out at any temperature on capillary columns by using as the input data retention times and peak widths at half height measured in isothermal runs. The mathematical model used for the calculation of the plate number was based on the Golay equation [5,6]. By using the retention times and the half height widths of the peaks obtained in isothermal runs it was possible to obtain the diffusion coefficients of the analysed compounds into the mobile and stationary phase and evaluate the column efficiency and the number of theoretical plates at any column temperature.

∗ Corresponding author. Tel.: +39 0103536113. E-mail address: [email protected] (P. Moretti). 0021-9673/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2009.10.059

The dependence on temperature and pressure of the calculated diffusion coefficient in the gas phase agreed with literature data [7,8], but the values of the Golay’s diffusion coefficient term in the stationary phase, Ds , have shown a maximum value at different temperatures for each compound, and decreased with increasing temperature after this point [4]. This fact is in contrast with the expected trend of the diffusion in solids, as the diffusion coefficient in polymers within different temperature ranges (40–200 ◦ C) increases with increasing temperature [9–12]. The observed behaviour suggests that the terms of the Golay equation contains some parameters depending on temperature. In fact, some authors who used the Golay equation were obliged to add corrective terms [13,14] or to modify the equation [1,15] in order to obtain predicted values corresponding with the experimental results. The efficiency of capillary columns can be expressed in terms of the effective plate height, heff that is the ratio between the adjusted retention time tR and the peak width at half height, wh , and is considered to be a better measure of the efficiency than h [16] as shown by Purnell [17]. As the tR is the product between the hold-up time tM and the retention factor k, one can suppose that the wh can be obtained with an empirical equation as the product of the tM and a new factor, kw , that has the same structure of the equation of k. Therefore, the effective plate height, obtained by the ratio between tR and wh , does not depend on the value of the tM but on k and kw only.

P. Moretti et al. / J. Chromatogr. A 1216 (2009) 8986–8991

2. Theory In isothermal and isobaric GC analysis on capillary columns, the height equivalent to one theoretical plate is deducted from the chromatogram with the following equation [16]: L h= 5.545

 w 2 h

where L is the column length, tR is the retention time and wh is the peak width at half height. In a previous paper [4], the prediction of the efficiency of gas chromatographic analysis by starting from the available experimental data was obtained by using the Golay equation [16] that correlates the theoretical plate height, h, in a small tract of length L along the column to the diffusion of the solutes in the gas and stationary phase, being the corresponding coefficients indicated with Dg and Ds , respectively: h=

2Dg + u¯



(1 + 6k + 11k2 )r 2 24Dg (1 + k)

2

+



2kdf2 3Ds (1 + k)

2

u¯

(2)

heff =

L 5.545

wh tR

2

(3)

where tR is the adjusted retention time: tR = tR − tM

(4)

correlated with the gas the hold-up time, tM , and with thermodynamic parameters by the equation [18–20]: tR = tM k = tM



Gs 1 exp − RT ˇ



(5)

where Gs is the free energy of solution or partitioning of a given compound between the gas and the liquid phase, R is the universal gas constant, T the absolute temperature and ˇ the phase ratio. If the wh can be expressed as a function of the gas the holdup time, than Eq. (3) can be simplified. In this work, the following empirical equation of wh is proposed [21]: wh = tM kw

(6)

where kw is a new quantity defined as: kw =





wh 1 Ew = exp − tM RT ˇ



(7)

In this way the Ew can be expressed as a function of temperature with a structure similar to that of Gs : Gs = Hs − TSs

T

H(T ) = H(T0 ) +



Cp dT

(9)

Cp dT T

(10)

T0 T

S(T ) = S(T0 ) + T0

where T0 is the lowest temperature of the selected range, and H(T0 ) and S(T0 ) are the enthalpy and entropy of solution at T0 . It was found by Fulem et al. [22] and Gonzalez [23] that the Cp depends linearly on temperature up to approximately 400–450 K. As the range of the experimental data used in this work is between 350.9 and 432.7 K, it is possible to accept a linear dependence of Cp on temperature and to calculate the values of free energy from experimental retention times by using the least squares method. Therefore, the structure of Ew becomes: Ew = A + BT + CT 2 + DT ln T

where k is the retention factor, r is the internal radius of the column, u¯ is the mean velocity of the carrier gas in L, df is the thickness of the stationary phase layer and Dg is the average value of the diffusion coefficient in the gas phase along L. With Eqs. (1) and (2) it is possible to calculate the values of Dg and Ds at any temperature and evaluate the column efficiency and the number of theoretical plates. However, this procedure is complex, depends on the uncertainty of the terms of the Golay equation (see Section 1) and causes valuable errors in the prediction of the column efficiency based on the Golay equation. Therefore, in this work a method is proposed for the prediction of the effective plate height, heff , that is considered to be a better measure of the efficiency than h [16,17]. Purnell [17] has shown that two peaks fully separated in a given set of experimental conditions (temperature, carrier gas flow rate) will be still resolved when the experimental condition change if the ratio between tR and peak width at the base, wb , (or between tR and wh ) and therefore the heff , is maintained constant, and not the ratio between tR and wb or wh :



where Hs and Ss refer to the enthalpy and entropy of solution respectively and can be expressed in term of the heat capacities, Cp :

(1)

tR

8987

(8)

(11)

where A, B, C and D are empirical constants that do not depend on temperatures. In this instance, the model that represents Ew is linear and therefore a unique exact solution is obtained and the calculation time is shorter. The efficiency of capillary columns in terms of the effective plate height, heff , can be calculated by using Eq. (3) instead of the Golay equation or of some of the suggested changes or additions to the original equation [1,13–15]. The method described is easier to be used with respect of the non-linear procedure based on the Eq. (2); in this case, taking into account Eqs. (5), (6) and (7), Eq. (3) can be rewritten as: heff =



L Ew − Gs exp −2 5.545 RT



(12)

The values of heff only depend on free energy of solution and on Ew . The results obtained in this work confirm this hypothesis. 3. Experimental Samples containing four members of each homologous series of straight-chain alkanes, 1-alcohols, 2-ketones and 1-alkenes (indicated for the sake of simplicity in the following text, in the tables and in the captions of the figures as C8, C10, . . .; C8OH, C9OH, . . .; C8K, C9K, . . .; C11E, C12E, . . ., respectively) obtained from Sigma–Aldrich (Steinheim, Germany) were used. Many analyses were carried out on a capillary column CP SIL 5CB coated with polydimethylsiloxane (PDMS) (Varian, Palo Alto, CA, USA). The column length was 30 m and the phase layer thickness 0.25 ␮m. The nominal internal diameter of the column was 0.320 mm, but the true value measured by scanning electron microscope using a Stereoscan 440 SEM, LEO, Cambridge, UK, was found to be 0.325 mm, and the true thickness of phase layer, also measured by SEM, was found to be equal to the nominal value within ±0.01 ␮m. In order to evaluate the effect of the phase layer thickness, some analyses were also carried out on a 30 m polydimethylsiloxane DB-1 column (Agilent Technologies, Wilmington, DE, USA), with nominal internal diameter of 0.320 mm and phase layer thickness of 3.0 ␮m. The actual values, checked by SEM, were 0.310 mm and 3.50 ␮m, respectively. The columns were installed in a Varian model 3800 gas chromatograph (Varian, Palo Alto, CA, USA) equipped with a split–splitless injector and a flame ionisation detector. Helium was used as the carrier gas. The split ratio was 1/20. The inlet pressure of the columns was measured with the electronic hardware of the gas chromatograph with an accuracy of ±0.1 psi (±689.5 Pa) and with a mercury manometer directly connected to the injector septum by

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P. Moretti et al. / J. Chromatogr. A 1216 (2009) 8986–8991

Fig. 1. Values of Gs /R (K) as a function of the column temperature (K) for two members of all the tested homologous series. Full lines: CP SIL 5CB column (30 m, I.D. 0.325 mm, layer 0.25 ␮m) at 173.3 kPa inlet pressure. Circles: n-alkanes C10 and C14, squares: 1-alcohols C8OH and C10OH, rhombs: 1-alkenes C11E and C13E, triangles: 2-ketones C9K and C11K (the black symbols refer to the compounds with less carbon atoms). Dashed lines: DB-1 column (30 m, I.D. 0.310 mm, layer 3.50 ␮m) 149.6 kPa inlet pressure. Black circle C10, black square C8OH.

means of a thin needle. The accuracy of these measurements was ±1 mmHg (±133.3 Pa). The column temperature, controlled by the gas chromatograph’s software with an approximation of ±1 ◦ C, was checked by a thermocouple with an accuracy of ±0.1 ◦ C. The barometric pressure was measured with the accuracy of ±0.1 mmHg (13.3 Pa) by using a precision mercury barometer and room temperature correction was applied. A Varian Star integration system was used for sampling the detector signal output at intervals of 0.06 s. The samples were injected as mixtures of pure compounds at the smallest amount permitted by the use of the micro syringe with the “needle tip” technique in order to obtain small, sharp and symmetrical peaks, near to the detection limit of the used detector and as close as possible to the infinite dilution condition. Thirty-six different analytical conditions were used for the CP SIL 5CB column, by coupling nine temperatures (350.9, 361.2, 371.4, 381.5, 391.6, 401.9, 412.2, 422.5 and 432.7 K) and four inlet pressures (52.8, 87.4, 121.8 and 173.3 kPa) for four members of each homologous series; seven different analytical conditions were used for the DB-1 column, by using seven temperatures (367.9, 378.3, 388.8, 399.2, 409.7 420.1 and 430.6 K) at 149.6 kPa inlet pressure for three members of straight-chain alkanes and 1-alcohols. Several runs were carried

Fig. 2. Values of Ew /R (K) as a function of the column temperature (K) for two members of all the tested homologous series. Full lines: CP SIL 5CB column (30 m, I.D. 0.325 mm, layer 0.25 ␮m) at 173.3 kPa inlet pressure. Circles: n-alkanes C10 and C14, squares: 1-alcohols C8OH and C10OH, rhombs: 1-alkenes C11E and C13E, triangles: 2-ketones C9K and C11K (the black symbols refer to the compounds with less carbon atoms). Dashed lines: DB-1 column (30 m, I.D. 0.310 mm, layer 3.50 ␮m) 149.6 kPa inlet pressure. Black circle C10, black square C8OH.

Fig. 3. Dependence of Gs /R (K) on temperature (K) for C12 (black circle) and C9OH (black square) at 121.8 and 173.3 kPa inlet pressure on the CP SIL 5CB column (dashed and dotted lines, respectively) and at 149.6 kPa inlet pressure for DB-1 column (full line).

out at each setting and the average values of the retention times and peak width calculated. The greatest fluctuation between replicate runs was of the order of few seconds for the retention time and 0.06 s for the peak width at half height, due to the sampling frequency of the integration system. The gas hold-up time tM at the different temperature and pressure tested was measured by injecting small amounts of methane. The experimental tM values were confirmed with the equation described previously [4–6] by using as input data the inlet and outlet pressure, the column length and diameter, the absolute temperature and the dynamic viscosity of the carrier gas. The results of the calculation agreed with the experimental values, showing the accuracy of the used parameters. 4. Results and discussion In order to evaluate the effect of temperature on the free energy Gs and on Ew and to check the validity of the experimental data, by using Eqs. (5) and (7) it is possible to estimate the Gs and Ew by starting from the experimental retention data. As an example, Figs. 1 and 2 show the values of Gs /R and Ew /R, respectively for two members of each of the four homologous series tested on the capillary column CP SIL 5CB at nine temperatures and at 173.3 kPa inlet pressure; and for the n-decane and 1-octanol on capillary column DB-1 at seven temperatures and at 149.6 kPa inlet pressure. Fig. 1 shows that the difference between the Gs /R on the two columns for both C10 and C8OH increases with increasing temperature (see

Fig. 4. Dependence of Ew /R (K) on temperature (K) for C12 (black circle) and C9OH (black square) at 121.8 and 173.3 kPa inlet pressure on the CP SIL 5CB column (dashed and dotted lines, respectively) and at 149.6 kPa inlet pressure for DB-1 column (full line).

P. Moretti et al. / J. Chromatogr. A 1216 (2009) 8986–8991

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Table 1 Relative percent error (E% = 100(heff,cal − heff,exp )/heff,exp ) between calculated and experimental values of the effective plate height, at four inlet pressure (52.8, 87.4, 121.8 and 173.3 kPa) and at nine temperatures (350.9, 361.2, 371.4, 381.5, 391.6, 401.9, 412.2, 422.5 and 432.7 K) on the CP SIL 5CB column. For sake of simplicity, in the text, in the tables and in the captions of the figures straight-chain alkanes are indicated as C8, C10, . . .; 1-alcohols as C7OH, C8OH, . . .; 2-ketones as C8K, C9K, . . .; 1-alkenes as C9E, C11E, . . .. T (K)

E(%)

P = 52.8 (kPa)

C8

C10

C12

C14

C9E

C11E

C12E

C13E

C7OH

C8OH

C9OH

C10OH

C8K

C9K

C10K

C11K

350.9 361.2 371.4 381.5 391.6 401.9 412.2 422.5 432.7

1.5 −3.5 1.0 1.4 1.3 −0.5 −2.9 2.0 −0.2

0.7 −2.3 1.5 1.0 0.2 −2.1 0.1 1.5 −0.6

0.7 −1.9 0.1 3.1 −1.0 −2.0 0.0 1.8 −0.7

−0.5 1.0 0.4 −1.0 −0.6 0.6 0.2 0.3 −0.3

1.4 −3.5 1.0 3.5 −1.3 −1.1 −1.1 1.7 −0.5

−1.8 3.3 −0.1 1.1 −4.3 −1.0 2.0 3.2 −2.3

0.1 −0.6 0.6 0.7 −1.1 −0.7 1.8 −0.9 0.1

−0.4 0.9 −0.4 0.1 0.0 −1.8 3.2 −1.7 0.3

−0.5 1.6 −1.9 2.2 −3.8 3.5 0.2 −1.7 0.6

0.7 −0.8 −2.1 3.9 −2.3 0.8 1.7 −3.2 1.4

0.3 −0.5 0.3 −1.4 1.6 0.2 1.0 −2.6 1.2

0.6 −1.0 0.0 0.0 −0.8 3.2 −0.6 −3.0 1.6

−0.9 2.0 −1.6 3.5 −4.4 0.5 −0.2 3.0 −1.6

−1.2 1.7 1.3 −0.4 −3.4 0.9 −0.2 3.3 −1.9

0.5 −0.9 0.3 −0.1 −0.4 2.7 −2.2 −0.2 0.5

0.1 −0.3 0.7 −0.7 −0.3 1.3 −0.4 −0.5 0.3

T (K)

E(%) C8

C10

C12

C14

C9E

C11E

C12E

C13E

C7OH

C8OH

C9OH

C10OH

C8K

C9K

C10K

C11K

−0.8 0.1 3.3 −1.0 −2.2 −3.9 6.7 −0.9 −0.9

0.3 −0.9 0.5 0.5 0.1 −1.5 1.9 −1.2 0.3

0.5 −0.4 −0.3 0.1 0.9 0.8 0.9 −1.1 1.2

−0.8 1.5 0.4 −1.2 −0.9 0.3 1.2 −0.2 −0.3

−0.2 −0.3 1.4 −0.1 −0.9 −2.2 3.5 −0.9 −0.2

0.6 −1.2 −0.2 0.6 0.8 −0.3 0.4 −1.5 0.8

0.3 −0.5 −0.3 0.1 0.6 0.1 0.4 −1.6 0.8

−0.1 0.4 0.0 −0.6 0.0 0.4 0.6 −1.1 0.4

0.0 −0.5 1.1 0.0 −0.4 −1.9 2.6 −0.6 −0.2

0.2 −0.7 0.5 0.2 0.0 −0.8 1.4 −1.1 0.3

0.3 −0.6 0.1 −0.1 0.4 0.1 0.6 −1.5 0.7

−0.3 0.5 0.5 −0.8 −0.5 0.1 1.0 −0.6 0.0

0.2 −0.8 0.6 0.3 −0.1 −1.4 1.9 −0.9 0.2

0.5 −0.9 0.0 0.4 0.4 −0.3 0.7 −1.4 0.7

0.4 −0.7 −0.3 0.1 0.7 0.3 0.3 −1.7 0.9

−0.1 0.2 0.1 −0.6 0.0 0.4 0.7 −1.2 0.4

P = 87.4 (kPa) 350.9 361.2 371.4 381.5 391.6 401.9 412.2 422.5 432.7 T (K)

E(%)

P = 121.8 (kPa)

C8

C10

C12

C14

C9E

C11E

C12E

C13E

C7OH

C8OH

C9OH

C10OH

C8K

C9K

C10K

C11K

350.9 361.2 371.4 381.5 391.6 401.9 412.2 422.5 432.7

1.0 −1.3 −1.2 0.0 0.9 3.3 0.0 −5.5 3.0

2.1 −3.1 −2.0 1.9 3.1 1.2 −1.3 −4.5 3.0

1.4 −1.8 −1.9 0.6 2.6 2.1 −0.8 −4.9 3.0

1.3 −1.4 −2.1 0.2 2.8 2.8 −2.2 −3.9 2.7

1.1 −1.8 −1.0 1.4 1.5 0.0 −0.4 −2.1 1.4

2.2 −3.1 −2.2 1.5 3.5 1.9 −1.7 −5.0 3.4

2.0 −2.6 −2.4 1.1 3.6 2.2 −1.9 −5.0 3.4

1.4 −1.7 −2.0 0.5 2.7 2.5 −1.3 −4.8 3.0

2.3 −3.4 −2.4 2.1 3.2 2.4 −2.1 −5.1 3.5

2.2 −3.2 −2.2 1.7 3.6 1.8 −2.2 −4.3 3.1

2.2 −3.1 −2.5 1.4 3.9 2.4 −2.4 −5.1 3.6

1.3 −1.6 −1.9 0.4 2.5 2.4 −1.2 −4.5 2.8

2.1 −3.1 −2.1 1.5 3.1 2.1 −1.2 −5.6 3.6

2.4 −3.5 −2.5 1.9 3.8 1.8 −1.7 −5.5 3.7

1.7 −2.2 −2.1 1.0 3.0 2.0 −2.0 −4.0 2.8

1.6 −2.0 −2.1 0.6 3.1 2.6 −2.0 −4.5 3.0

T (K)

E(%)

P = 173.3 (kPa)

C8

C10

C12

C14

C9E

C11E

C12E

C13E

C7OH

C8OH

C9OH

C10OH

C8K

C9K

C10K

C11K

−2.2 1.8 5.1 −2.1 −1.9 −4.6 −4.4 5.8 −6.5

−1.2 3.0 −0.5 −2.8 1.9 −0.1 −1.7 2.8 −1.2

−0.2 0.1 0.8 −1.5 1.9 −2.0 0.5 0.9 −0.5

0.1 −0.6 0.3 1.2 −0.2 −1.9 −0.1 2.2 −1.0

−1.2 1.8 1.9 −4.1 2.9 −0.4 −5.7 6.1 −3.4

−0.3 −1.1 3.0 1.4 −3.4 −1.7 −0.8 6.3 −3.0

0.0 0.2 1.1 −3.6 2.6 1.1 −0.4 −1.7 0.9

−0.5 0.8 1.0 −2.3 0.6 0.4 0.0 0.2 −0.2

−1.7 2.2 3.7 −5.1 0.8 −0.3 −3.7 5.3 −3.6

0.3 −2.4 3.9 −0.8 −0.7 −0.6 −3.1 6.0 −2.4

−1.7 3.9 −0.3 −3.2 0.8 0.3 −0.1 1.6 −1.0

−1.2 1.3 2.8 −1.8 −2.7 0.3 −0.5 4.4 −2.4

−1.0 0.3 4.4 −2.7 −1.4 −1.9 0.3 5.2 −2.8

0.3 0.3 −2.9 3.5 −1.1 0.8 −1.2 0.1 0.3

−0.1 −0.6 1.1 1.3 −1.2 −1.4 −2.2 5.6 −2.3

−1.2 1.9 0.8 0.6 −3.7 −0.5 1.2 3.0 −2.0

350.9 361.2 371.4 381.5 391.6 401.9 412.2 422.5 432.7

full and dashed lines with circle and square symbols). This means that the values of the retention factor, k, and of the kw on the DB1 column with 3.50 ␮m phase thickness column decreases faster with increasing temperature than on the CP SIL 5CB column. This behaviour is more evident for the Ew /R values shown in Fig. 2 and therefore the effect of temperature on the kw values is greater. In Figs. 3 and 4 the dependence of Gs /R and Ew /R on temperature is plotted for C12 and C9OH at 121.8 and 173.3 kPa on the CP SIL 5CB column and at 149.6 kPa on the DB-1 column. The trend of Gs /R of C12 (Fig. 3) is similar to that of C9OH at the same pressure on the 0.25 ␮m CP SIL 5CB column (dashed lines). It changes with changing the phase thickness on the 3.5 ␮m DB-1 column (full lines). The same behaviour is observed for the Ew /R values (Fig. 4) but, whereas the Gs /R values only slightly change with changing the inlet pressure of the column, the variation of the Ew /R and of the effective plate height, heff , and the difference between the two

columns are greater. As the behaviours are similar, the model suggested in Eqs. (6) and (7) for the prediction of the peak width at half height as a function of the dead time and of kw leads to accurate results, as the relative percent errors range between +0.4% and −0.6% for adjusted retention times and between +2.3% and −2.3% for the wh values. In Figs. 5 and 6 the dependence of heff on temperature for the members of each of the four homologous series tested on the CP SIL 5CB column at nine temperatures and at 173.3 kPa inlet pressure is shown. Two figures with different heff range on the vertical axis are shown, in order to permit a better resolution of the various plots. Fig. 7 shows the heff values for C10 and C8OH on the DB-1 column at seven temperatures and at 149.6 kPa inlet pressure. In all these figures, the symbols indicate the experimental values, while the full lines plots show the trend of heff obtained by placing into Eq. (3) the values of tR and wh calculated with Eqs. (5), (6) and (7).

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Fig. 5. Dependence of heff (mm) on temperature (K) for members of the four homologous series tested on the CP SIL 5CB column at nine temperatures and at 173.3 kPa inlet pressure. The heff scale ranges between 0.8 and 2.2 mm. Circle: C14; rhomb: C13E; square: C10OH; triangle: C11K. The symbols show the experimental values and the lines show the trend of the heff calculated with Eqs. (3), (5), (6) and (7). See text. Table 2 Relative percent error (E% = 100(heff,cal − heff,exp )/heff,exp) between calculated and experimental values of the effective plate height, at 149.6 kPa inlet pressure and at sevent temperatures (367.9, 378.3, 388.8, 399.2, 409.7 420.1 and 430.6 K) on the DB-1 column. T (K)

E(%)

P = 149.6 (kPa)

C8

C10

C12

C7OH

C8OH

C9OH

367.9 378.3 388.8 399.2 409.7 420.1 430.5

0.1 −0.5 0.4 0.1 −0.3 0.1 0.0

−0.2 0.5 −0.4 0.5 −1.2 1.2 −0.4

−0.1 0.1 0.0 0.2 −0.7 0.8 −0.2

−0.3 0.8 0.0 −0.9 −0.1 1.0 −0.4

0.3 −0.9 0.7 0.4 −0.6 0.0 0.1

−0.1 0.8 −1.3 0.5 1.1 −1.2 0.4

Tables 1 and 2 show, for all the experimental runs carried out, the relative percent errors, E(%) = 100(heff,cal − heff,exp )/heff,exp , between calculated and experimental values of the effective plate height of Eq. (3), for the CP SIL 5CB and DB-1 columns respectively. The maximum percent error ranges between +6.7% and −6.5% for CP SIL 5CB and between +1.2% and −1.3% for DB-1. The greatest errors for the column with the smaller phase thickness are due to the structure of Eq. (3), because if the ratio between the calculated adjusted retention time and the experimental one is smaller

Fig. 6. Dependence of heff (mm) on temperature (K) for members of the four homologous series tested on the CP SIL 5CB column at nine temperatures and at 173.3 kPa inlet pressure. The heff scale ranges between 0.0 and 12.0 mm. Circle: C10; rhomb: C11E; square: C8OH; triangle: C9K. The symbols show the experimental values and the lines show the trend of the heff calculated with Eqs. (3), (5), (6) and (7). See text.

Fig. 7. Dependence of heff (mm) on temperature (K) for the C10 and C8OH tested on the DB-1 column at seven temperatures and at 149.6 kPa inlet pressure. The heff scale ranges between 0.6 and 1.6 mm. Circle: C10; square: C8OH. The symbols show the experimental values and the lines show the trend of the heff calculated with Eqs. (3), (5), (6) and (7). See text.

(or greater) than unity and the ratio between the calculated and experimental peak width at half height is smaller (or greater) than unity, the values of E(%) increase. In fact, if the measure of the experimental effective plate height can be expressed as: heff = heff,o ± heff

(13)

where heff,o is the height equivalent to the effective theoretical plate calculated with the Eq. (3), then it is possible to calculate the value of the indetermination of the heff value, heff , as [2]:



heff =

∂heff ∂wh

2

 wh2

o

+

∂heff ∂tR

2

 tR2

o

+

∂heff ∂tM

2 2 tM o

(14)

∼ 0.06 s and L = 3000 cm, then the indeif wh ∼ = 0.06 s; tR = tM = termination heff in the measurement can be approximated as: heff ∼ = 0.12

heff,o wh

(15)

In this instance, the greatest value of the heff found was 16.3 mm for the most volatile compound (n-C8) at the highest pressure and temperature. This corresponds to 33.3% in term of relative percent error. In Fig. 8 the comparison of heff for C12 at 121.8 and 173.3 kPa on the CP SIL 5CB and at 149.6 kPa on the DB-1 column on temperature

Fig. 8. Comparison of the dependence on temperature (K) of heff (mm) for C12 at 121.8 and 173.3 kPa inlet pressure on the CP SIL 5CB (dashed and dotted lines respectively) and at 149.6 kPa inlet pressure on the DB-1 column (full line). The heff scale ranges between 0.0 and 3.5 mm. The symbols show the experimental values and the lines show the trend of the heff calculated with Eqs. (3), (5), (6) and (7). See text.

P. Moretti et al. / J. Chromatogr. A 1216 (2009) 8986–8991

8991

the values of heff . The structure of the equation that defines kw (Eq. (7)) is similar to the definition of retention factor, k, therefore the method for computing the values of free energy from experimental retention times and the values of the Ew from experimental peaks width at half height by using the least squares method is identical, the time spent for these computations is small and the model employs linear parameters leading to an unique exact solution. On the contrary, the models based on the original Golay equation or on its more complex modifications, are not linear in the parameters, and do not yield an unique solution. References

Fig. 9. Comparison of the dependence on temperature (K) of heff (mm) for C9OH at 121.8 and 173.3 kPa inlet pressure on the CP SIL 5CB (dashed and dotted lines respectively) and at 149.6 kPa inlet pressure on the DB-1 column (full line). The heff scale ranges between 0.0 and 3.5 mm. The symbols show the experimental values and the lines show the trend of the heff calculated with Eqs. (3), (5), (6) and (7). See text.

is shown. Fig. 9 shows the behaviour of C9OH in the same analytical conditions of Fig. 8. By changing the inlet pressure of the column at constant temperature the heff and the Ew /R values change (see the distance between the dotted and the dashed line). Therefore, whereas the adjusted retention time changes with inlet pressure due to the change of the velocity of the carrier gas, the peak width at half height depends also on the change of the Ew . The full line shows that on the DB-1 column with a ticker stationary phase layer, the heff values less depend on temperature. 5. Conclusions If the peak width at half height is expressed as a function of the gas hold-up time and of the new quantity, kw , this model shows that Ew depends on the inlet pressure of the columns (whereas Gs remains about constant) and how this dependence influences

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