Temperature dependence of the Kováts retention index

Journal of Chromatography A, 1206 (2008) 178–185

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Temperature dependence of the Kováts retention index Convex or concave curves Miklós Görgényi a,∗ , Zoltán A. Fekete a , Herman Van Langenhove b , Jo Dewulf b a b

Department of Physical Chemistry, University of Szeged, Rerrich Béla tér 1, H-6721 Szeged, Hungary Research Group ENVOC, Faculty of Bioscience Engineering, Ghent University, Coupure Links 653, B-9000 Ghent, Belgium

a r t i c l e

i n f o

Article history: Received 26 June 2008 Received in revised form 3 August 2008 Accepted 4 August 2008 Available online 7 August 2008 On the 50th anniversary of the Kovats retention index. Keywords: Kováts retention index Temperature dependence of the retention index Alcohols Solvation heat capacity Polyethylene glycol Convex or concave curves

a b s t r a c t The non-linearity in the temperature dependence of the Kováts index, I (the formation of convex or concave curves) was characterized by the second derivative, d2 I/dT2 . The expression deduced on a purely mathematical–physicochemical basis is d2 I/dT2 = [2TS(CH2 )dI/dT − 100ıCp ]/TG(CH2 ). The solute-dependent factor for dI/dT, d2 I/dT2 , and the extreme temperature in the I vs. T relationship is ıCp , which is the molar solvation heat capacity difference between the solute and a hypothetical n-alkane which elutes at the same time as the given solute, while the solventdependent factors are the solvation entropy and free energy of the methylene unit, S(CH2 ) and G(CH2 ). Experimentally, convex I vs. T curves with a minimum are formed when ıCp  0, while concave ones with a maximum are observed when ıCp  0. In the event of a linear temperature dependence, the former equation can be simplified: dI/dT = 100ıCp /2TS(CH2 ). The deviation from linearity (higher d2 I/dT2 ) increases with increasing ıCp values. The model equations were tested from the dataset published by the Kováts group on C78 (19,24-dioctadecyldotetracontane), POH (18,23-dioctadecyl-1-untetracontenol), PCN (1-cyano-18,23-dioctadecyluntetracontane) and TMO (1,38dimethoxy-17,22-bis-(16-methoxyhexadecyl)-octatriacontane) and by present measurements on the Innowax phase. © 2008 Elsevier B.V. All rights reserved.

1. Introduction

existence of an extreme in the I vs. T function, if a simple linear [4,5]:

The concept of the Kováts retention index, I, was introduced 50 years ago [1]. I can be defined by the corrected retention times, the retention volumes or the free energies of solvation, G, of the solute, x, and the bracketing n-alkanes with carbon numbers z and z + 1: Gx − Gz I = 100 + 100z Gz+1 − Gz

(1)

0021-9673/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.chroma.2008.08.006

(2)

or a logarithmic model is used [6]: I =A+

The interest of researchers in the retention index continues nowadays: publications on new retention indices of interesting new classes of compounds are continuously determined, or discussions on the Kováts indices are published [2,3]. For identification purposes, I values at different temperatures are often necessary. The temperature dependence is almost exclusively characterized by the experimentally determined index increments, dI/dT, or from I values determined at some other temperature(s). The determination of I at only a few temperatures may conceal up the

∗ Corresponding author. Tel.: +36 62 544628; fax: ++36 62 544652. E-mail address: [email protected] (M. Görgényi).

I = a + bT

B T +C

(3)

Taking various signs of the parameters A, B and C in Eq. (3), there are eight distinct possibilities. It was shown by the analysis of Tudor and Moldovan [7], that in fact only two cases occur for the temperature dependence of the retention index: the curve is either convex, i.e. increasing I with T curves (B < 0), or concave, i.e. decreasing I with T curves (B > 0). A is always positive and C is always negative. The retention of more polar compounds (e.g. alcohols [8], thiols [9], ketones [10], nitroalkanes and cyano compounds [11]) on apolar phases leads to an extreme in the I vs. T relationship. Some observations indicate that the I vs. T curve may have a maximum (e.g. 1,3-diketones on DB-5 [12], or dialkyl ethers on a Carbowax phase [13]). Some semiquantitative results predict that a minimum is related to increased molar solvation heat capacities of the solutes compared to n-alkanes [11].

M. Görgényi et al. / J. Chromatogr. A 1206 (2008) 178–185

179

The thermodynamic deduction of Eq. (3) [14], assumes the condition that both the solvation enthalpy (H < 0) and entropy (S < 0) are temperature-independent, which significantly reduces the applied temperature range. Eqs. (2) and (3) can be used only when the retention index changes with temperature without an extreme. Reddy et al. [15], suggested a quadratic expression for the temperature dependence of the retention index:

To test the various models described here, we selected the large dataset of Kováts group [15,23–25] on C78, POH, PCN and TMO phases and our own experimentally determined retention and thermodynamic data on an Innowax column. Cluster and principal component analysis on C78 phases demonstrated that the C78, POH, PCN and TMO phases furnish significant information for a description of the retention in the C78 solvent family [2].

I(T ) = I(T ◦ ) + AT T + ATT T 2

2. Theory

(4)

In Eq. (4) T = T − T◦ , AT and ATT are parameters determined from the experimental retention index data, and I(T◦ ) is the retention index at the reference temperature, T◦ = 403 K. A combined logarithmic equation may be applicable in almost every system in a wide temperature range [10,16]: I =A+

B + C ln T T

ıG(x−z) G(CH2 )

(6)

Sx − Sz + 100z Sz+1 − Sz

I (S) was similarly defined as I in Eq. (6) [17] I  (S) = 0.01I(S) − z =

ıS(x−z)

(10)

S(CH2 )

where the ıS functions are the solvation entropy differences of the solute, x (Sx ), and the n-alkanes eluting before (carbon number z) (Sz ). For the derivation of Eq. (9), we used the following fundamental thermodynamic equations [26]:



∂G ∂T





= −S and P

∂S ∂T



= P

Cp T

d dT

 S(CH )  2

G(CH2 )

(Cp (CH2 )/T )G(CH2 ) + S(CH2 )S(CH2 )

=

G2 (CH2 ) (11)

(7)

where S(CH2 ) and G(CH2 ) are the appropriate solvation thermodynamic functions of the methylene unit. The entropy index, I(S), in Eq. (7) is defined similarly to the Kováts index: I(S) = 100

(9)

and basic differentiation rules:

We recently introduced the first derivative of the retention index with respect to temperature, dI/dT [17]: S(CH2 ) dI = (I − I(S)) dT G(CH2 )

S(CH2 )  dI  = (I − I  (S)) dT G(CH2 )

(5)

Calculation of retention indices at a given temperature via Eqs. (2)–(5) requires a knowledge of the parameters, which can be determined empirically. To achieve a non-empirical thermodynamics based interpretation of the temperature dependence of the retention index (Eq. (1)), its differentiation with respect to temperature would be necessary. This is possible only from a corrected form, I , of the retention index [17]: I  = 0.01I − z =

To obtain the second temperature derivative of I (or I , Eq. (6)) we recall the first derivative of I from [17]:

d   d I (S) = dT dT =

(8)

The application of Eq. (7) requires knowledge of the thermodynamic quantities at different temperatures, which can be calculated from the molar heat capacities. Unfortunately, solvation molar heat capacities are determined only infrequently, although they are very informative in characterizing solvation interactions [18]. AndreoliBall et al. concluded that a large negative excess heat capacity, CpE is observed when a structure is destroyed (e.g. long chain-alkanes in globular molecules). A positive CpE is associated with structure formation in a solution of an alcohol in n-alkane, for instance [19,20]. The main problem in the determination of Cp is that it needs retention measurements over a rather large temperature interval (T ≥ 100 K), and possibly many more data points than typically measured [21]. The I vs. T plot is more or less curved with a positive or negative slope. No investigation appears to have been performed as to why it is convex or concave. Along the I vs. T convex curves, the slope of the curve, tan ˛ regularly increases in value, and therefore d2 I/dT2 > 0, while for concave curves tan ˛ regularly decreases, and therefore d2 I/dT2 < 0 [22]. The value of d2 I/dT2 is a measure of the curvature of the I vs. T plot. The main aim of the present work is twofold: to introduce the second temperature derivative of the retention index (this is the first attempt to give a thermodynamic explanation of I vs. T curve types), and to validate this thermodynamic model with experimental data from the literature and our own GC data.



ıS(x−z)



S(CH2 )



(ıCp,(x−z) /T )S(CH2 ) − ıS(x−z) Cp (CH2 )/T



S 2 (CH2 ) (12)

We take Eq. (9) as a product in the second differentiation: d2 I  dT 2

=

(Cp (CH2 )/T )G(CH2 ) + S(CH2 )S(CH2 ) S(CH2 ) + G(CH2 )



2 G(CH2 )

(I  − I  (S))

S(CH2 )  (I − I  (S))− G(CH2 )

(ıCp,(x−z) /T )S(CH2 ) − ıS(x−z) (Cp (CH2 )/T )



2 S(CH2 )

(13)

Multiplication, rearrangement, simplification and substitution lead to d2 I  1 = G(CH2 ) dT 2



ıCp,(x−z) Cp (CH2 )  dI  − I + 2S(CH2 ) T T dT



(14) We have defined a hypothetical n-alkane with carbon number  which elutes between the n-alkanes with carbon numbers z and z + 1 and at the same time as the solute x [17]. Its solvation molar heat capacity, Cp, , is Cp, = Cp (CH2 )I  + Cp,z

(15)

180

M. Görgényi et al. / J. Chromatogr. A 1206 (2008) 178–185

Therefore, ıCp = Cp,x − Cp,

(16)

From Eq. (6): dI 

= 0.01

dT

dI dT

(17)

and 2 

d I d2 I = 0.01 2 dT dT

(18)

Substitution of Eqs. (15)–(18) into Eq. (14) results in 2

d I 1 = TG(CH2 ) dT 2

 2TS(CH2 )

dI − 100ıCp dT

 (19)

In the event of a linear temperature dependence, dI/dT = const. and d2 I/dT2 = 0. Applying this condition to Eq. (19) and rearranging, we obtain a simple thermodynamic equation: 100ıCp dI = dT 2TS(CH2 )

(20)

If ıCp = 0, dI/dT = 0, the solute behaves as a hypothetical nalkane, in agreement with the original definition of the Kováts retention index [1]. 3. Experimental Eq. (20) was tested on some representative solutes of a very large dataset [15,23–25] on C78 (19,24-dioctadecyldotetracontane), POH (18,23-dioctadecyl-1-untetracontenol), PCN (1-cyano-18,23dioctadecyluntetracontane) and TMO (1,38-dimethoxy-17,22-bis(16-methoxyhexadecyl)-octatriacontane). In their dataset, the Kováts group established a linear temperature dependence for the retention of the solutes (ATT = 0 in Eq. (4)), with the exception of alcohols on the C78 and POH phases. The standard molar enthalpy, entropy and heat capacity of solvation are given at 403 K. The method of their determination is outlined in [27]. The enthalpy and entropy data at other temperatures were calculated with the help of molar solvation heat capacities. Both the Kováts group and ourselves used a temperature-independent heat capacity model. However, some references suggest that in a large temperature range it cannot be considered fully temperature-independent [28]; e.g Cp of nbutanol in dimethylpolysiloxane]. The dataset does not provide examples for a non-linear temperature dependence without an axtreme in the I vs. T graph. Therefore, to test Eq. (19), we measured the retention indices and thermodynamic functions of some compounds with different functionalities (nitromethane, chloroform, isopentyl acetate, 1-hexanol and 1heptanol) on the highly polar HP Innowax fused silica open tubular column (0.32 mm, film thickness 0.5 ␮m, 30 m). The temperature range was wide, 45–150 (160) ◦ C. No extension of the temperature range was possible for the applied solutes. All of the solutes exhibit a non-linear temperature dependence of the retention index. The gas chromatographic measurements were performed on a 5890 Hewlett-Packard Series II gas chromatograph with a flame ionization detection system. The retention data were evaluated with the HP 3365 ChemStation. The carrier gas was nitrogen. The gas hold-up time, tM , was determined by methane injection into the sample mixture. The net retention volume, VN , was calculated from the equation [29] VN = (tr − tM )jFa

Tc Pa − Pw Ta Pa

(21)

where tr is the retention time, j is the compressibility factor, Fa is the flow rate at ambient temperature, Ta , Tc is the column temperature, Pa is the barometric pressure, and Pw is the saturated water vapour pressure. As some retention times were quite long (several hours), attention was paid to making the measurements under stable meteorological conditions (at approximately constant barometric pressure). The solvation thermodynamic functions H, S and Cp were determined by parameter estimation [24], from ln

HTo STo + R ln w Cp VN =− + − T RT R R

T − T

o

T

− ln

T To



(22)

where  = 83.14 cm3 bar mol−1 K−1 , R = 8.314 J K−1 mol−1 , from the g-SPOT-s (standard chemical potential difference of the solute with the ideal gas phase reference) related to the molal Henry coefficient [29], g (in kg bar mol−1 ) at the reference temperature T◦ = 373 K, and w is the mass of the stationary phase in kg. w is not known exactly for the capillary column used. However, it is not necessary, because for the calculation we require only the value of S(CH2 ). It can be calculated from the differences between S (=STo + R ln w [27]) of neighboring n-alkanes. G(CH2 ) can also be calculated also exactly from H and S(CH2 ) data. Some representative values of S(CH2 ) and G(CH2 ) are published in Ref. [17], however, they may be slightly different from the values used in the calculations as S(CH2 ) depends somewhat on the n-alkane pairs. None of these solutes on the Innowax column exhibited linear temperature dependence of I. dI/dT and d2 I/dT2 at different temperatures were determined from the derived Eq. (5):

 dI  dT

=−

B C + T T2

(23)

At the extreme temperature is the T, where the I vs. T graph has a minimum or maximum. At this point dI/dT = 0 Textr =



d2 I dT 2

B C

(24)



=

2B C − 2 T3 T

(25)

4. Results and discussion 4.1. Linear temperature dependence of the retention index (Eq. (20)) Theoretically the simplest case is when the I vs. T plot is linear (Eq. (2)). However, this condition is rarely fulfilled exactly; some deviation from linearity always occurs. Reddy et al. considered this deviation not significant for the applied solutes except alcohols on the C78 and POH phase [15]. The linearity model was tested on the retention and thermodynamic data on 30 different types of solutes on C78, POH, PCN and TMO phases. We have now included the data on the C78 and POH phases (Tables 1 and 2). Similar calculations were performed on the retention and thermodynamic data on the PCN and TMO phase. Fig. 1 shows that the experimental and calculated (Eq. (20)) dI/dT data are in very good agreement with each other on the four phases: dI dI (calc) = (0.0029 ± 0.00112) + (0.948 ± 0.003) (exp) dT dT n = 120, r = 0.948

(26)

The relative percentage error of the estimation, 100[dI/dT(exp) − dI/dT(calc)]/dI/dT(exp), is 8.0 ± 4.5% (C78 phase), 6.6 ± 4.7% (POH), 5.1 ± 7.7% (PCN) and 6.9 ± 10.6% (TMO). In case of linear temperature dependence, the quantities in the bracket of

M. Görgényi et al. / J. Chromatogr. A 1206 (2008) 178–185

181

Table 1 Testing the linear model (Eq. (20)) at 403 K on the C78 phase, based on the experimental data of Kováts group [15,25]

2-Butanone 2,2-Dimethylbutane Hexafluorobenzene 1-Hexene 1-Cyanopropane Hexamethyldisiloxane 2-Hexyne Pentanal 1-Fluorohexane 1-Nitropropane Fluorobenzene 1,4-Dioxane Benzene 2,3-Dimethylpentane 1-Butanethiol Pyridine 1-Acetoxybutane 2,3,4-Trimethylpentane Cycloheptane 1-Chlorohexane Dibuty lether Ethylbenzene Anisole 1,1,1-Trifluorodecane 2,3-Dimethylpyridine Bromobenzene cis-Hydrindane Nitrobenzene trans-Decalin Naphthalene

dI/dTa

dI/dTb

2TS(CH2 )dI/dTc

100ıCp d

d2 I/dT2 × 105 e

0.074 0.091 −0.214 0.055 0.117 −0.246 −0.022 0.135 0.008 0.175 0.223 0.246 0.257 0.097 0.318 0.344 −0.053 0.129 0.414 0.165 0.006 0.303 0.307 −0.225 0.327 0.534 0.592 0.559 0.656 0.789

0.065 0.081 −0.190 0.050 0.105 −0.199 −0.021 0.123 0.008 0.159 0.204 0.226 0.235 0.087 0.294 0.319 −0.051 0.119 0.389 0.154 0.006 0.286 0.294 −0.212 0.310 0.509 0.565 0.534 0.632 0.761

−289.9 −356.5 838.3 −215.4 −458.3 963.6 88.1 −540.8 −32.0 −701.0 −893.3 −985.4 −1029.5 −388.6 −1294.4 −1400.2 215.7 −525.1 −1698.5 −676.9 −24.6 −1243.1 −1266.9 928.5 −1349.4 −2203.7 −2457.3 −2320.4 −2728.3 −3294.1

−253.6 −318.8 745.2 −194.9 −411.4 778.6 83.6 −491.5 −32.6 −636.6 −817.1 −904.9 −940.2 −350.5 −1196.1 −1300.1 206.9 −483.8 −1595.0 −632.1 −23.4 −1173.0 −1213.0 873.7 −1280.1 −2100.0 −2345.0 −2216.0 −2629.4 −3177.4

3.9 4.1 −10.1 2.2 5.1 −20.0 −0.5 5.5 −0.1 7.2 8.5 9.0 10.0 4.3 11.2 11.4 −1.0 4.7 12.0 5.2 0.1 8.1 6.3 −6.4 8.1 12.1 13.2 12.3 11.7 13.9

a

Experimental value. Linear model for the temperature dependence of I. Calculated from Eq. (20). c Using experimental dI/dT. d Calculated via Eqs. (15) and (16) from experimental Cp values. e Calculated from Eq. (19). G(CH2 ) and S(CH2 ) values were calculated from G and S of the bracketing n-alkanes. G(CH2 ) and S(CH2 ) values were published in [17]. b

Eq. (19) should be equal. This good agreement is indicated by the similarity of the data in the columns 4 and 5 of Tables 1 and 2. The calculated data somewhat overestimate the dI/dT values, probably because the I vs. T plots are not completely straight lines. d2 I/dT2 , the measure of the curvature of the I vs. T plot, is low for the solutes with the exceptions of hexafluorobenzene, trifluorodecane and hexamethyldisiloxane on the C78 and POH phases. These are compounds with a considerable negative temperature dependence in I.

In our recent paper [17], Eq. (20) was not known. Merely an excellent linear plot of dI/dT vs. ıCp on C78 was given for the solutes with the exception of the alcohols, without mathematical derivation, on the basis of the thermodynamics and experimental observations. The present Eq. (20) proves that the forecast was correct. The published value of the slope of the dI/dT vs. ıCp plot at 403 was −0.0257 ± 0.00021 [17]. The present calculations furnish 100/2TS(CH2 ) = −0.0243 ± 0.00035 on C78 (Eq. (20)). Eq. (20) contains a solute-dependent (ıCp ) and a solventdependent (S(CH2 )) term. On a given phase, therefore, dI/dT is a function only of the heat capacity difference term, i.e. how the Cp values of the solute and the hypothetical n-alkane differ due to structural change during the solvation. S(CH2 ) is always negative, and therefore the sign of dI/dT for a linear model (in a narrow temperature range) is determined by ıCp . From Eq. (20): dI >0 dT

if ıCp < 0

(27)

if ıCp > 0

(28)

and dI <0 dT

Fig. 1. Check on the validity of Eq. (20). Comparison of the calculated dI/dT values with the experimental values on four phases: C78 ( ), POH (䊉), PCN (), TMO ().

If we compare the dI/dT and ıCp values in the Tables 1 and 2, the general correctness of the relations in Eqs. (27) and (28) are validated. For thermodynamic and mathematical reasons, the opposite cases in Eqs. (27) and (28) (e.g. dI/dT > 0 if ıCp > 0) are not possible.

182

M. Görgényi et al. / J. Chromatogr. A 1206 (2008) 178–185

Table 2 Testing the linear model (Eq. (20)) at 403 K on the POH phase based on the experimental data of Kováts group [15,25]

2-Butanone 2,2-Dimethylbutane Hexafluorobenzene 1-Hexene 1-Cyanopropane Hexamethyldisiloxane 2-Hexyne Pentanal 1-Fluorohexane 1-Nitropropane Fluorobenzene 1,4-Dioxane Benzene 2,3-Dimethylpentane 1-Butanethiol Pyridine 1-Acetoxybutane 2,3,4-Trimethylpentane Cycloheptane 1-Chlorohexane Dibutyl ether Ethylbenzene Anisole 1,1,1-Trifluorodecane 2,3-Dimethylpyridine Bromobenzene cis-Hydrindane Nitrobenzene trans-Decalin Naphthalene

dI/dTa

dI/dTb

2TS(CH2 )dI/dTc

100ıCp d

d2 I/dT2 × 105 e

−0.040 0.114 −0.244 0.038 0.058 −0.196 −0.006 −0.066 0.018 0.195 0.262 0.161 0.335 0.122 0.213 0.019 −0.117 0.160 0.427 0.220 −0.044 0.359 0.300 −0.247 −0.022 0.610 0.626 0.567 0.682 0.694

−0.035 0.102 −0.221 0.036 0.056 −0.162 −0.005 −0.061 0.017 0.187 0.249 0.153 0.340 0.103 0.197 0.016 −0.113 0.149 0.405 0.208 −0.041 0.357 0.288 −0.234 −0.020 0.601 0.602 0.557 0.661 0.668

157.7 −449.3 961.7 −149.8 −228.6 772.5 24.3 267.6 −73.0 −790.6 −1062.2 −633.1 −1358.2 −494.6 −877.3 −78.3 481.9 −659.0 −1772.4 −913.2 182.6 −1490.2 −1254.9 1033.2 92.0 −2551.7 −2633.8 −2385.6 −2874.9 −2942.3

137.9 −403.3 869.4 −141.6 −218.9 639.9 20.1 246.3 −70.3 −756.4 −1011.2 −622.2 −1380.1 −417.1 −812.3 −67.3 464.5 −614.6 −1679.5 −862.7 171.7 −1481.0 −1205.6 977.5 84.0 −2512.5 −2534.8 −2343.3 −2788.0 −2833.8

−1.9 4.5 −9.1 0.8 1.2 −13.0 −0.5 −2.7 0.3 3.9 6.5 4.7 -2.5 9.8 7.5 1.3 −2.0 5.1 10.9 5.9 −1.3 1.1 5.8 −6.6 −1.0 4.6 11.8 5.0 10.4 13.1

a

Experimental value. Linear model for the temperature dependence of I. Calculated from Eq. (20). c Using experimental dI/dT. d Calculated via Eqs. (15) and (16) from experimental Cp values. e Calculated from Eq. (19). G(CH2 ) and S(CH2 ) values were calculated from G and S’ of the bracketing n-alkanes. G(CH2 ) and S(CH2 ) values at some temperatures were published in [17]. b

4.2. Non-linear temperature dependence of the retention index (Eq. (19)) It is inevitable from the d2 I/dT2 values (Tables 1 and 2) that there is a slight non-linearity in the I vs. T plot for the solutes in Tables 1 and 2, although a constant dI/dT (linear model) has been suggested for them. On use of the non-linear model Eq. (19) for the model compounds on the four phases, excellent linearity is observed between d2 I/dT2 and the difference term in the brackets in Eq. (19) (Fig. 2). The curvature of the I vs. T plots of alcohols on the C78 and POH (and Innowax) phases varies to a much higher extent than for other solutes (Table 3). Accordingly, we made another plot to represent the linearity between d2 I/dT2 and the difference 2TS(CH2 )dI/dT − 100ıCp (Eq. (19)) for the alcohols on C78 and POH, and also for other solutes on the Innowax phase (Fig. 3). The slope of the plot is higher for the Innowax phase because of the higher G(CH2 ). G(CH2 ) characterizes the polarity of the stationary phase [30,31]. The difference in the various C78 phases is only in the functional groups at the end of the long alkane chain. The calculated G(CH2 ) values are therefore not significantly different for the C78, POH, PCN and TMO phases: the average value is −2131 ± 14 J mol−1 . For comparison purposes, Table 3 shows ıCp calculated in two different ways: either from Eqs. (15) and (16) or expressed from rearranging Eq. (19). Similar values are obtained from both calculations for this set of data. It must be stressed, however, that the latter way is preferred considering the effect of measurement errors. Since Eq. (16) contains the difference of values of similar magnitude, catastrophic loss of significant digits occurs, when the

Cp data are contaminated with considerable uncertainty (which is typical, see e.g. our data on polar column in Table 4). The propagation of errors is less severe for the case of Eq. (19), as demonstrated later in the past column of Table 5. Figs. 2 and 3 demonstrate that, at a given temperature, the more positive the difference term in the brackets in Eq. (19) (i.e. the more negative ıCp is), the more negative d2 I/dT2 is. This is the case for concave curves. Correspondingly, the more positive ıCp is, the larger d2 I/dT2 is (convex curves).

Fig. 2. Check on the validity of Eq. (19) for solutes with an almost linear temperature dependence of the retention index (low d2 I/dT2 value). Symbols as in Fig. 1.

M. Görgényi et al. / J. Chromatogr. A 1206 (2008) 178–185

183

Table 3 Testing the validity of the Eq. (19) at 403 K on the C78 and POH phases, based on the experimental data of Kováts group [15,25]. d2 I/dT2 × 105 b

dI/dTa

Phase

d2 I/dT2 × 105 c

ıCp d

ıCp e

Textr f

C78

2-Methyl-2-butanol 2-Pentanol 2-Methyl-2-pentanol 1-Pentanol 2-Hexanol 1-Hexanol 1-Heptanol

0.12 0.04 0.01 0.09 0.06 0.11 0.12

273 132 224 147 76 147 94

300 140 240 160 80 160 100

20.5 10.1 15.9 9.6 4.3 8.4 5.2

23.0 10.8 17.3 10.7 4.6 9.5 5.7

363 372 360g 347g 328g 399 332g

POH

2-Methyl-2-butanol 2-Pentanol 2-Methyl-2-pentanol 1-Pentanol 2-Hexanol 1-Hexanol 1-Heptanol

−0.38 −0.44 −0.47 −0.55 −0.43 −0.55 −0.55

954 1023 925 1187 1074 1237 1115

920 1120 1060 1300 1180 1360 1260

90.5 107.2 100.0 126.0 109.8 128.8 117.3

87.8 115.6 111.6 135.8 118.9 139.3 129.5

471 450 458 459 449 460 462

a

Calculated from the first temperature derivative of Eq. (4). From Eq. (19). c Calculated from the second temperature derivative of Eq. (4). d Calculated via Eqs. (15) and (16), in J K−1 mol−1 . e From Eq. (19). G(CH2 ) and S(CH2 ) values were calculated from G and S of the bracketing n-alkanes. For G(CH2 ) and S(CH2 ) values, see Tables 1 and 2. dI/dT and d2 I/dT2 data are experimental values. f Extreme temperature, Textr was calculated from Eq. (4). g Below the experimental temperature range. b

Table 4 Parameters of Eq. (5) and thermodynamic functions of the solutes calculated on the Innowax column (based on experimental data from this work) Solute

A

B

C

Ha

S’b

Cp b

Isopentyl acetate Chloroform Nitromethane 1-Hexanol 1-Heptanol Decane Undecane Dodecane Tridecane Tetradecane Pentadecane

405.4 ± 34.6 −634.4 ± 113.4 −2134 ± 132.8 2948 ± 105 3171 ± 116

26491 ± 1831 80189 ± 5987 145899 ± 7010 −73766 ± 5558 −84156 ± 6099

112.9 ± 5.0 247.1 ± 16.4 495.4 ± 19.2 −232.6 ± 15.2 −247.9 ± 16.7

−39.2 ± 0.1 −35.7 ± 0.1 −39.6 ± 0.2 −48.4 ± 0.1 −51.8 ± 0.1 −35.0 ± 0.1 −38.4 ± 0.1 −41.8 ± 0.1 −45.3 ± 0.1 −48.8 ± 0.1 −52.3 ± 0.1

−184.4 ± 0.3 −179.6 ± 0.3 −183.5 ± 0.4 −198.3 ± 0.3 −202.7 ± 0.3 −179.8 ± 0.3 −184.2 ± 0.3 −188.8 ± 0.3 −193.6 ± 0.3 −198.2 ± 0.3 −203.0 ± 0.3

57.2 ± 7.3 57.5 ± 5.9 69.9 ± 10 73.4 ± 7.5 76.9 ± 7.6 49.6 ± 8.1 56.0 ± 7.3 63.9 ± 7.4 70.9 ± 7.3 77.8 ± 7.4 84.9 ± 7.5

a b

In kJ mol−1 . In J mol−1 K−1 . S contains an unknown correction term Rlnwsp ,where wsp is the mass of the stationary phase [27].

Table 5 Estimation of ıCp values from Eq. (19) on the Innowax column (based on experimental data from this work)a

Isopentyl acetate Isopentyl acetate Nitromethane Nitromethane Chloroform Chloroform Chloroform Chloroform 1-Hexanol 1-Hexanol 1-Hexanol 1-Hexanol 1-Heptanol 1-Heptanol 1-Heptanol 1-Heptanol a

T (K)

I

dI/dTb

d2 I/dT2 × 103 d

323 373 323 373 323 373 403 324c 323 373 403 317c 323 373 403 339c

1139.7 1145.0 1179.9 1190.7 1041.5 1043.8 1046.9 1041.5 1375.7 1372.9 1369.0 1375.8 1478.2 1477.4 1475.0 1478.5

0.10 ± 0.02 0.11 ± 0.02 0.14 ± 0.09 0.28 ± 0.07 0.0 0.09 ± 0.06 0.12 ± 0.05 0.0 −0.01 ± 0.07 −0.09 ± 0.06 −0.12 ± 0.05 0.0 0.04 ± 0.08 −0.06 ± 0.06 −0.10 ± 0.06 0.0

0.5 0.2 3.9 2.1 2.4 1.3 0.9 2.4 −2.2 −1.2 −0.8 −2.3 −2.6 −1.5 −1.1 −2.2

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.1 0.1 0.5 0.3 0.4 0.3 0.2 0.4 0.4 0.2 0.2 0.4 0.4 0.3 0.2 0.3

ıCp e 0 −3 ± 1 20 ± 4 4±3 15 ± 4 5±3 2±2 15 ± 2 −13 ± 3 −4 ± 2 −1 ± 2 −15 ± 2 −18 ± 4 −7 ± 3 −3 ± 2 −14 ± 2

Standard errors are estimated from the propagation of error throughout. Calculated via Eq. (23) from the retention index. c Textreme values. Calculated via Eq. (24). Textreme (isopentyl acetate) = 234 K, Textreme (nitromethane) = 294 K, both calculated data are far out of the GC temperature range. d Calculated via Eq. (25) from the retention index. e Calculated from Eq. (19) using experimental thermodynamic data. G(CH2 ) = −1984 ± 3 J mol−1 (323 K), −1725 ± 4 J mol−1 (373 K), −1593 ± 4 J mol−1 (403 K) or G(CH2 ) = (−664.7 ± 12.3)·e−(2943±55)/RT . −1679 J mol−1 (393 K) was reported on Carbowax 20 M [32]. S(CH2 ) = −5.69 ± 0.03 J mol−1 K−1 (323 K), −4.66 ± 0.02 J mol−1 K−1 (373 K), −4.12 ± 0.02 J mol−1 K−1 (403 K). S(CH2 ) and G(CH2 ) were determined from G and S values of decane–tetradecane. G(CH2 ) = (−1.153 ± 0.056)·e−(4313±146)/RT . b

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Fig. 3. Test of Eq. (19) for solutes with high d2 I/dT2 value. Alcohols (see Table 3) on C78 ( ) or POH (䊉), and also other solutes (see Table 5) on the Innowax phase ().

The extreme temperature, Textr can be determined from Eq. (24) or starting from Eq. (19) and applying the condition dI/dT = 0: Textr = −

100ıCp G(CH2 )d2 I/dT 2

The parameters of Eq. (5) and the thermodynamic functions of solvation on the Innowax phase used for further calculations are shown in Table 4. The characteristics of the temperature dependence of the retention index, the d2 I/dT2 and ıCp values of solvation, are listed in Table 5. At the minimum temperature (convex case, d2 I/dT2 > 0), ıCp is definitely positive, while at the maximum temperature, ıCp is negative (Table 5). An increased excess solvation heat capacity is connected with the formation of structure in any solvent. This suggestion is supported by the observations that alcohols may form micelles (e.g. tetramers) in the apolar stationary phase [19,20]. In the concave case (e.g. alcohols on polar phase), the opposite process, i.e. destruction of structure, is probable. The explanation requires further investigations. ıCp for solutes with any type of curve approaches zero with increasing temperature (Table 5), i.e. the interactions with the stationary phase show even less specificity relative to the n-alkanes. S(CH2 ) is more positive on the highly polar Innowax phase, than on the C78 phases (Tables 1 and 5). In a short temperature range, Eq. (20) can be applied, the dI/dT increasing with decreasing S(CH2 ). Therefore, the lower negative value of S(CH2 ) leads to a higher dI/dT:dI/dT of a solute is higher on polar phases than on an apolar one.

(29)

This equation demonstrates that Textr is the higher, the larger ıCp is (Tables 3 and 5; see for alcohols), while it is the lower, the higher the column polarity is. The calculations should be performed at the extreme temperature. Eq. (29) is another simple equation with which to calculate ıCp (at Textr ). Eqs. (5), (19) and (29) forecast that an extreme is a fundamental mathematical characteristic of an I vs. T curve, although the calculated value may often be outside the normal GC range. The solutes selected for measurements on the Innowax column provide excellent examples (e.g. I vs. T curves with a maximum and a minimum) for the testing of Eq. (19). Isopentyl acetate and nitromethane have distinctly convex I vs. T curves with dI/dT > 0 on the Innowax phase. Chloroform has a minimum (the minimum temperature is around 50 ◦ C), while 1-hexanol and 1-heptanol have a maximum (the maximum temperature is around 60 ◦ C) in the I vs. T plot. The temperature dependence of the retention indices of solutes with an extreme in the I vs. T relationship (chloroform and 1-heptanol) is shown in Fig. 4.

5. Conclusions The second temperature derivative of the retention index can be expressed with a simple equation. This equation rationalizes the occurrence of concave or convex curves. On a given phase, the curve type, the extent of curvature and even the extreme temperature are controlled by the sign and difference in the molar solvation heat capacity of the solute and a hypothetical reference n-alkane, ıCp . The stationary phase-dependent factors are G(CH2 ) and S(CH2 ). ıCp can be determined by the definition equations, but its determination may be more exact in some cases if we use the stationary-phase dependent thermodynamic and retention factors. The deviation from linearity increases with increasing ıCp , the curve becoming more and more convex or concave. Convexity is associated to a more negative ıCp , and concavity with a more positive ıCp . The validity of the equation describing dI/dT and d2 I/dT2 was checked by several calculations. The theoretical conclusions are in agreement with the experimental data of the Kováts group and with our measurements on Innowax. Acknowledgements The authors thank Professor Ervin sz. Kováts for supplying the basic experimental data. M.G. is grateful for support from the Department of Physical Chemistry of the University of Szeged (OTKA K72989) and the Special Research Fund of Ghent University (VBO 173, 2006). References

Fig. 4. Temperature dependence of the retention indices of chloroform (×) and 1heptanol ( ) on the Innowax stationary phase.

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