Extending the approach of the temperature-dependent potential to the small alkanes CH4, C2H6, C3H8, n-C4H10, i-C4H10, n-C5H12, C(CH3)4, and chlorine, Cl2

Chemical Physics 298 (2004) 195–203 www.elsevier.com/locate/chemphys

Extending the approach of the temperature-dependent potential to the small alkanes CH4, C2H6, C3H8, n-C4H10, i-C4H10, n-C5H12, C(CH3)4, and chlorine, Cl2 Uwe Hohm a

a,*

, Lydia Zarkova

b

€ Physikalische und Theoretische Chemie der Technischen Universit€at Braunschweig, Hans-Sommer-Straße 10, Institut fur D-38106 Braunschweig, Germany b Institute of Electronics, Blvd Tzarigradsko Chaussee 72, 1784 Sofia, Bulgaria Received 21 October 2003; accepted 26 November 2003

Abstract The concept of the temperature dependence of the binary intermolecular interaction potential is extended to anisotropic molecules. The effective Lennard-Jones (n
6) potentials with explicitly temperature-dependent potential parameters eðT Þ (potential welldepth) and Rm ðT Þ (separation at minimum energy) are successfully applied to the alkanes Cn H2n þ 2 (n ¼ 1–5), and Cl2 . The potential parameters eðT Þ, Rm ðT Þ and n (repulsive parameter) are determined by simultaneously fitting thermophysical equilibrium (second pVT and acoustic virial coefficients, BðT Þ and bðT Þ, respectively) and transport data (viscosity gðT Þ and self-diffusion coefficients .DðT Þ). For these molecules it is shown that an effective isotropic temperature-dependent potential reproduces the experimental input data for pure gases and their binary mixtures within the range of the stated experimental accuracy. The root-mean-square deviations between experimental and calculated data are by 7%–70% smaller using the Lennard-Jones (n
6) temperature-dependent potential compared to its temperature-independent analogue. Correlations are found for the potential parameters with the volume of the molecules, their dispersion interaction energies and the enlargement of the molecular size on account of vibrational excitation. Ó 2003 Elsevier B.V. All rights reserved.

1. Introduction The determination of intermolecular interaction potentials is one of the central problems in physical chemistry. Once the binary and multi-particle interaction energies are known as a function of the space-coordinates a variety of thermophysical properties, such as densities, viscosities, diffusion-coefficients, condensation rates, etc., can be calculated for the fluid (gas and liquid) state. These calculations become more and more important as the experimental investigations sometimes are time consuming, laborious, dangerous and expensive. In order to describe the interactions between two molecules a vast number of approaches are available today. Starting from the very simple hard-core potential one can end up with * Corresponding author. Tel.: +49-0-5313915350; fax: +49-05313914577. E-mail addresses: [email protected] (U. Hohm), lzarkova@ yahoo.com (L. Zarkova).

0301-0104/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2003.11.026

the very accurate ab initio potentials. However, in the same direction the computational work to obtain the interaction energies increases enormously. High-level ab initio potentials are still limited to rather small molecules. Usually the intermolecular interaction potential U ðR; nA ; vA ; nB ; vB ; xa ; xB Þ is supposed to be independent of temperature T . R is the distance of molecules A and B, their mutual orientation is denoted by xA and xB , whereas n and v are the electronic and vibrational quantum states. However, as already outlined by Hirschfelder et al. [1] a temperature-dependent form of the potential can be obtained from a Boltzmann averaging of the intermolecular potential over the different orientations and/or over different quantum states, U ðR; T Þ  hU ðR; nA ; vA ; nB ; vB ; xa ; xB Þin;v;x . The explicit dependence of U on different quantum states or the explicit use of a temperature-dependent potential is, however, still limited to some exceptional cases (see e.g. [2–5]). In 1992 Stefanov [6] and later on Zarkova [7] have started to systematically investigate effective isotropic

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U. Hohm, L. Zarkova / Chemical Physics 298 (2004) 195–203

temperature-dependent intermolecular interaction potentials (ITDPs). They have used a Lennard-Jones (n
6) type potential with explicitly temperature-dependent parameters eðT Þ and Rm ðT Þ. In the following this is called a Lennard-Jones (n
6) temperature-dependent potential, LJTDP. It was shown that the LJTDP is able to reproduce and predict thermophysical properties of the neat globular gases CF4 , CCl4 SiF4 , SiCl4 SF6 , MoF6 WF6 , UF6 C(CH3 )4 , Si(CH3 )4 , and the quasi-globular molecule BF3 with very high accuracy [8]. Additionally, thermophysical properties of binary mixtures of these molecules and also with the noble gases could be calculated with low error bounds in a wide range of temperature [9,10]. In the present paper we extend the concept of the LJTDP to non-spherical molecules. We have chosen a series of alkanes (methane, ethane, propane, n-butane, iso-butane, n-pentane, and neopentane) and chlorine. In the case of these molecules the intermolecular interaction energies have been described before by multicenter Lennard-Jones (12
6) potentials (see e.g. [11–16]). We show that within our concept it is possible to use much simpler isotropic one-center potentials if the temperature dependence is taken into account by suitable models. Additionally, we show that simple Lorenz– Berthelot mixture rules applied to the potential parameters of the LJTDP allow for a precise calculation of low-pressure thermophysical properties of binary mixtures of alkanes. We have to note that our aim is to reproduce thermophysical properties of gases caused by binary interactions of freely rotating molecules. If these molecules are trapped in confined geometries (e.g. condensed phase or surface), however, interaction energies as a function of the separation and different relative orientations must be taken into account. Ab initio intermolecular interaction potentials of this kind are available in the recent literature for small alkanes [17–19].

2. Theory The isotropic temperature-dependent potential chosen in this work is a Lennard-Jones (n
6) potential (LJTDP) of the form "
n
6 # eðT Þ Rm ðT Þ Rm ðT Þ U ðR; T Þ ¼ 6
n : ð1Þ n
6 R R Here R is the intermolecular distance, eðT Þ is the effective minimum energy of the potential at R ¼ Rm ðT Þ, and n is the repulsive parameter, which does not depend on the temperature in this model. We consider the temperature dependence of Rm ðT Þ and eðT Þ to result from the vibrational excitation of the molecule. In the case of the LJTDP a simple model to account for this excitation is

used. First we restrict ourselves to harmonic vibrations only. If we take a0k to be the corresponding classical vibrational amplitude of frequency xk in the lowest vibrational level with v ¼ 0, then the relative change Cv of the amplitude in higher vibrational levels with respect to the first excited level is given by [20] pffiffiffiffiffiffiffiffiffiffiffiffiffi 2v þ 1
1 : ð2Þ Cv ¼ pffiffiffi 3
1 By summing over all excited states the thermally averaged change dðT Þ of the vibrational amplitudes between two interacting molecules is now modeled as N N Y 1 X S2;k S1;‘ ; Zvib k¼1 ‘¼1; ‘6¼k
 1 X
j
hx‘ ¼ Dðj; g‘ Þ exp ; kB T j¼1
 1 X
j
hx‘ ¼ C‘ Dðj; g‘ Þ exp ; kB T j¼1

dðT Þ ¼ 2d0  S1;‘ S2;‘

ð3Þ

Zvib is the vibrational partition function of N harmonic oscillators with frequency x‘ and ground-state degeneracy g‘ . Dðj; g‘ Þ is the degeneracy of a harmonic oscillator excited to the jth level. d0 can be regarded as the change of the classical amplitude between the first excited level and the ground state averaged over all vibrational modes. Beside Rm , e, and n, d0 now is the fourth parameter of the LJTDP. The temperature dependence of Rm ðT Þ and eðT Þ is then given by Rm ðT Þ ¼ Rm ð0Þ þ dðT Þ;
eðT Þ ¼ eð0Þ

Rm ð0Þ Rm ðT Þ

ð4Þ

6 :

ð5Þ

Eq. (5) follows from the reasonable assumption that the long-range dispersion energy does not depend on the temperature [21]. In the case of mixtures the following simple mixing rules are used: n12 ¼ ðn1 þ n2 Þ=2; Rm12 ðT Þ ¼ ½Rm1 ðT Þ þ Rm2 ðT Þ=2; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e12 ðT Þ ¼ e1 ðT Þe2 ðT Þ:

ð6Þ ð7Þ ð8Þ

Here, the indices Ô1Õ and Ô2Õ denote the individual molecules, whereas Ô12Õ characterizes the mixture.

3. Determination of the LJTDP parameters The potential parameters Rm ð0Þ, eð0Þ, n, and d0 of the LJTDP are obtained by minimizing the sum F of squared deviations F ¼

M X i¼1

R2i

¼

M  X i¼1


 2  Pcalc;i    ln  aexp;i ; Pexp;i 

ð9Þ

U. Hohm, L. Zarkova / Chemical Physics 298 (2004) 195–203

RMS ¼

pffiffiffiffiffiffiffiffiffiffiffi F =M ;

ð10Þ

between M experimental (Pexp ) and calculated values (Pcalc ) of the second pVT and acoustic virial coefficients, BðT Þ and bðT Þ, and viscosity, gðT Þ. In the case of methane, self-diffusion coefficients qDðT Þ are also taken into account. The deviations are normalized to the accepted relative errors aexp ¼ DPexp =Pexp of the measurements, RMS denotes the root-mean-square (rms) error of the fit. There is a long discussion about the error criterion (see the recent work of [22]). We have shown in many cases that Eq. (9) is a suitable criterion for the determination of the optimal potential parameters (see [8] and references therein). In the case of small deviations ðPcalc
Pexp Þ=Pexp , Eq. (9) is equivalent to the maximum likelihood method. The optimal parameters are determined by looking for the global minimum of F ½eð0Þ; Rm ð0Þ; n; d0 . The determination of the optimal parameters is not a simple and straightforward procedure. The experimental input datasets have to be checked for consistency, their error bounds must be analyzed critically and the influence of each experiment on the resulting parameters must be studied. These laborious and time-consuming procedure is described in detail by Zarkova and Hohm [8] and is not repeated here.

Table 1 Experimental input data for determination of the potential parameters Substance

Experimental input data

References

CH4

85 13 34 10

BðT Þ, 110–623 K bðT Þ, 200–350 K gðT Þ, 200–510 K qDðT Þ, 154–359 K

[36] [40,41,43] [42,44–47] [48]

C2 H6

216 BðT Þ, 191–623 K 8 bðT Þ, 223–351 K 37 gðT Þ, 293–633 K

[36] [43] [42,44,49–52]

C3 H8

147 BðT Þ, 211–623 K 24 bðT Þ, 225–375 K 54 gðT Þ, 296–478 K

[36] [39,43,53] [42,44,49,50,54–56]

n-C4 H10

144 BðT Þ, 244–588 K 8 bðT Þ, 250–320 K 38 gðT Þ, 293–626 K

[36] [57] [42,44,49,50,58,59]

i-C4 H10

29 BðT Þ, 251–573 K 8 bðT Þ, 200–350 K 34 gðT Þ, 223–627 K

[36] [60] [46,58,61,62]

n-C5 H12

84 BðT Þ, 260–648 K 7 bðT Þ, 270–330 K 49 gðT Þ, 293–548 K

[36] [38] [69]

C(CH3 )4

34 BðT Þ, 270–548 K 8 bðT Þ, 250–323 K 34 gðT Þ, 263–473 K

[36] [63] [63,69]

Cl2

22 BðT Þ, 244–1079 K 13 bðT Þ, 260–440 K 18 gðT Þ, 233–772 K

[12,36,64] [35] [65–68]

197

References to the experimental input data as well as the corresponding temperature ranges are given in Table 1. Note that the primary data given in [36] rather than their smoothed values are used throughout this work. The temperature dependence of dðT Þ is calculated via the vibrational partition function ZðT Þ of the harmonic oscillator. The normal vibrational frequencies xk are taken from the compilation of Sverdlov et al. [23]. 4. Results and discussion In this study we have applied the concept of an isotropic potential to (optically) isotropic as well as nonisotropic molecules. We display the aspect-ratio of the molecules via the ratio r ¼ Da=a, where Da is the polarizability anisotropy and a the mean dipole-polarizability of the molecule. In the case of CH4 and C(CH3 )4 we have r ¼ 0, whereas for C2 H6 , C3 H8 , n-C4 H10 , iC4 H10 , n-C5 H12 and Cl2 r is 0.15, 0.17, 0.21, 0.14, 0.23, and 0.57, respectively [24]. The numbers show that the alkanes exhibit a moderate electro-optic anisotropy. On the other hand, the anisotropy of the shape might be somewhat different from these numbers. 4.1. Pure gases By using the LJTDP we are able to calculate simultaneously thermophysical equilibrium and transport data. In order to visualize this successful behaviour deviation plots are given for the pure gases in Figs. 1–8. Standard methods are used for the calculation of the thermophysical properties [1]. The final parameters of the LJTDP at T ¼0 K are summarized in Table 2. For comparison we have also determined the temperatureindependent Lennard-Jones (n
6) parameters, which result from setting d0  0 in Eq. (4). The accuracy with which the experimental data are reproduced is given by the rms-error RMS, Eq. (10), in the last column. Rm ðT Þ can be represented with very high accuracy by Rm ðT Þ ¼ Rm ð0Þ þ a1 expð
b1 =T Þ þ a2 expð
b2 =T Þ: ð11Þ The fit parameters are given in Table 3. eðT Þ can be calculated with Eq. (5), the repulsive parameter n is given in Table 2. This enables the use of the LJTDP at any other temperature in the range DT given in Table 3. 4.1.1. The n-alkanes Due to the comparably high vibrational frequencies of methane, this molecule does not show any noticeable vibrational excitation in the temperature range of T < 650 K considered in this work. As a result the exponential terms in Eq. (3) are practically zero. Therefore, due to the scattering of the experimental input data, d0 cannot be determined in our minimization

U. Hohm, L. Zarkova / Chemical Physics 298 (2004) 195–203

D P /a / % exp

2

2

B

CH4

b h rD

0

D P /a / % exp

198

0

-2

-2

200

200

400

T/K

Fig. 1. Normalized deviations Ri between the LJTDP solution (zero line) and the experimental input data of CH4 .

∆ P /aexp / %

β η

2 0 -2

B b h

1 0 -1 -2

400

T/K

Fig. 2. Normalized deviations Ri between the LJTDP solution (zero line) and the experimental input data of C2 H6 .

C 3H 8

4

300

600

B b h

2 0

400

500 600 T/K

700

Fig. 6. Normalized deviations Ri between the LJTDP solution (zero line) and the experimental input data of n-C5 H12 .

DP /a / % exp

200

B b h

2 1 0 -1

-2

C(CH3)4

-2

-4

-3

400

T/K

Fig. 3. Normalized deviations Ri between the LJTDP solution (zero line) and the experimental input data of C3 H8 .

4

B b h

n-C4H10

2 0 -2 -4 200

300

600

400

500 T / K

Fig. 7. Normalized deviations Ri between the LJTDP solution (zero line) and the experimental input data of C(CH3 )4 .

4 D P /a / % exp

200

DP /a / % exp

600

n-C5H12

2

-4

D P /a / % exp

T/K

Fig. 5. Normalized deviations Ri between the LJTDP solution (zero line) and the experimental input data of i-C4 H10 .

B

C2H 6

4

400

600

D P /a / % exp

-4

B b h

i-C4H10

Cl2

2

B b h

0 -2 -4

400

T/K

600

Fig. 4. Normalized deviations Ri between the LJTDP solution (zero line) and the experimental input data of n-C4 H10 .

procedure. Any reasonable choice of d0 will result in dðT Þ  0. Consequently, we choose d0  0, although the physically correct enlargement is clearly d0 > 0. Hence,

200

400

600

800

1000 T/K

Fig. 8. Normalized deviations Ri between the LJTDP solution (zero line) and the experimental input data of Cl2 .

the LJTDP reduces to a usual Lennard-Jones (n
6) potential, which is abbreviated LJ afterwards. The deviation plot for methane, CH4 , is shown in Fig. 1. The

U. Hohm, L. Zarkova / Chemical Physics 298 (2004) 195–203

199

Table 2 Parameters of the LJTDP at T ¼ 0 K and Lennard-Jones (n
6) potential with temperature-independent parameters (LJ) Substance

Rm ð0Þ (10
10 m)

e=k (K)

n

d0 (10
10 m)

RMS

CH4 a

3.868  0.002

220.78  0.33

21.63  0.18

0

0.997

C2 H6

LJTDP LJ

4.447  0.001 4.307  0.001

364.18  0.35 425.5  0.22

22.28  0.05 38.29  0.08

0.0241  0.004 –

1.487 2.361

C3 H8

LJTDP LJ

4.930  0.003 4.774  0.001

478.38  0.35 580.39  0.18

24.12  0.21 71.09  0.19

0.0168  0.0001 –

1.174 1.788

n-C4 H10

LJTDP LJ

5.284  0.001 5.090  0.009

595.80  0.31 771.5  6.1

21.10  0.04 264  30

0.0195  0.0001 –

1.393 2.459

i-C4 H10

LJTDP LJ

5.363  0.004 5.479  0.001

554.42  0.44 482.90  0.24

22.72  0.08 21.88  0.06

0.0165  0.0001 –

0.866 3.040

n-C5 H12

LJTDP LJ

5.495  0.002 5.366  0.001

765.2  2.2 921.0  1.1

30.48  0.87 949.74  0.39

0.0121  0.0001 –

0.663 1.115

C(CH3 )4

LJTDP LJ

5.697  0.004 5.630  0.004

619.02  0.80 688.30  0.46

37.28  0.29 100.4  1.4

0.007  0.0003 –

0.958 1.030

Cl2

LJTDP LJ

4.259  0.010 4.248  0.009

499.4  5.5 506.7  4.1

26.34  0.80 27.89  0.53

0.0088  0.0003 –

1.473 1.484

a

In this case, LJTDP and LJ are identical because of d0  0, see text.

Table 3 Coefficients for Rm ðT Þ according to Eq. (11) Substance

a1 (10
10 m)

b1 (K)

a2 (10
10 m)

b2 (K)

Tmin –Tmax (K)a

CH4 C2 H6 C3 H8 n-C4 H10 i-C4 H10 n-C5 H12 C(CH3 )4 Cl2

0 0.119997 0.185499 0.326082 0.264736 0.313346 0.180731 0.031754

–

0 0.881982 1.09974 1.78472 1.38971 1.57872 0.747022 0

– 2149.996 1952.022 1758.367 1644.370 1844.566 1963.417 –

100–900 180–700 200–700 225–700 250–700 250–700 225–600 225–1100

a

605.470 451.278 339.383 282.247 332.084 443.890 1095.689

Suggested temperature range for the validity of the LJTDP.

deviations shown are normalized to their individual experimental error. In the temperature range considered in this work quantum effects need not be taken into account. The BðT Þ and bðT Þ deviations are situated symmetrically towards the zero-line, which represents the properties calculated with the LJTDP. In the case of qDðT Þ and gðT Þ we found a weak systematic deviation. However, most of the deviations are lying in the range between 1.5 experimental error aexp . The potential parameters Rm ð0Þ and eð0Þ of the present study are in excellent agreement with Rm ð0Þ ¼ 3:867  10
10 m and eð0Þ=k ¼ 217 K of a numerical potential obtained by Matthews and Smith [25]. Ethane is the first of the non-isotropic molecules in the alkane series. We observe a considerable improvement of the calculated thermophysical properties of the LJTDP compared to the LJ potential (see Table 2). As can be seen in Fig. 2 the deviations of the experimental input data are situated symmetrically around the LJTDP solution. In the case of propane the LJTDP yields an essential improvement compared to the isotropic LJ potential. The deviations for the individual properties are shown in

Fig. 3. They are distributed symmetrically towards the LJTDP solution. The parameters of the LJTDP can be compared with potential parameters obtained through the inversion of acoustic virial coefficients bðT Þ [26]. According to the representation of the interaction energy in terms of a Maitland–Smith function with distance-dependent repulsive parameter nðRÞ, Trusler et al. [26] obtained Rm ¼ 4:8620  10
10 m and e=k ¼ 560:56 K. Their result for Rm is in good agreement with our LJTDP parameters, whereas a moderate deviation is found for e=k. In the case of the two butanes the rms-error in Table 2 increases strongly from 1.393 (0.866) to 2.459 (3.040) for n-butane (iso-butane) when going from the temperaturedependent LJTDP to the temperature-independent LJ. The deviation plots are shown in Figs. 4 and 5. For both butanes a symmetric deviation between experimental and calculated properties can be seen. The failure of the LJ potential was also observed by Seok and Oxtoby [27]. They were not able to fit the experimentally observed nucleation rate of n-butane when applying a temperature-independent Lennard-Jones potential. Fig. 6 shows the deviation plot for n-pentane. The LJTDP reproduces the experimental data within their

200

U. Hohm, L. Zarkova / Chemical Physics 298 (2004) 195–203

stated accuracy. This reproducibility within the errorbounds is also observed by using the LJ potential, although the rms-error is higher by 90%. Neopentane is a spherical molecule. The LJTDP works slightly better than the LJ model. The nearly symmetrically situated deviations are presented in Fig. 7. 4.1.2. Chlorine In the present study we have also considered chlorine because of its quite large anisotropy, its simple structure and its very well-defined spectroscopic properties. This allows for a thorough comparison of its LJTDP potential parameters with other molecular quantities. However, despite of its industrial importance only very few experimental thermophysical input data are available. Amongst them there are the very precise measurements of the speed-of-sound in low pressure gaseous chlorine [35]. Our approach does not yield a significant improvement of the LJTDP compared to the simpler Lennard-Jones (n
6) potential. The experimental data are reproduced nearly within their uncertainty, see Fig. 8. A similar behaviour of the intermolecular interactions was also observed by Hurly [35], who was able to reproduce the experimental data of bðT Þ and BðT Þ with a spherical symmetric hard-core Lennard-Jones (12
6) potential with Rm ¼ 4:152  10
10 m and e=k ¼ 531:02 K. These characteristic quantities are quite close to our own findings Rm ¼ 4:259  10
10 m and e=k ¼ 499:52 K and cast some doubt on the two-center Lennard-Jones model proposed by Bohn et al. [12].

agreement of 1.5% between the experimental data of gmix ðT Þ and our calculations. This is demonstrated in Fig. 9 for binary methane–propane mixtures with different compositions. The experimental results for B12 ðT Þ of different binary mixtures with n-butane are compared in Fig. 10. Overall an agreement of less than 5% can be observed. In the present work we have not invoked the properties of the binary mixtures into the minimization procedure which yields the potential parameters of the pure gases. Considering binary mixtures we have restricted ourselves to the simple Lorenz–Berthelot mixing rules. On account of the large error of the very few available experimental mixture properties it is hardly possible to test more complex mixing rules. 4.3. Interpretation of the LJTDP parameters We have to remind that the LJTDP approach presented here yields an effective isotropic intermolecular interaction potential. This potential model is successfully used to calculate thermophysical properties of the neat gases and of some of their binary mixtures. We have not taken their anisotropy into account. This means that the parameters of the LJTDP might be influenced to some extent by the anisotropy of the mole-

4.2. Binary mixtures

Fig. 9. Relative deviations between experimental and calculated mixed viscosity data gmix ðT Þ of binary mixtures of methane and propane with different compositions. The experimental data measured by different authors are taken from [69] (closed symbols) and [58] (open symbols).

D B12 / %

Bzowski et al. [28] were the first to present a systematic investigation on the thermophysical properties of gaseous mixtures. They have shown for a number of small polyatomic gases and five noble gases that the low density viscosities gmix ðT Þ, interaction B12 ðT Þ-virial and diffusion coefficients D, and the thermal diffusion factors can be reproduced from the potentials of the pure gases by using suitable mixing-rules. The deviations are 1% for gmix ðT Þ, 10–15% for B12 ðT Þ, and 5% for D12 ðT Þ. In the present study we compare only the first two quantities with available measurements. The LJTDP parameters for the unlike interactions are obtained by Eqs. (6)–(8). Despite of their simplicity these mixture rules allow for a reliable reproduction of the scarcely available experimental thermophysical data. This has already been demonstrated for binary mixtures between globular gases [9,10]. We have calculated the mixture viscosity gmix ðT Þ and the interaction virial coefficient B12 ðT Þ for all mixtures, where we have found experimental data to compare with. In general, the pVT interaction virial coefficient B12 ðT Þ is measured with lower accuracy than the viscosity of a mixture gmix ðT Þ. In all cases we observe an

6

C2H6 - n-C4H10 C3H8 - n-C4H10

4

i-C4H10 - n-C4H10

2 0 -2 -4 300

350

400

T/K

450

Fig. 10. Relative deviations between experimental and calculated second virial coefficients B12 ðT Þ of binary mixtures with n-butane. Data were taken from [37].

U. Hohm, L. Zarkova / Chemical Physics 298 (2004) 195–203

n-C4H10

5.0 4.5 CH4

C 3H8

3.5 3.0 3.5

i-C4H10

Although the main concern is with the temperaturedependent Lennard-Jones (n
6) potential, LJTDP, a remark should be addressed to the behaviour of the repulsive parameter n of the LJ potential, given in Table 2. In the case of the unbranched alkanes n is increasing with the size of the molecule. Starting from n ¼ 21:63

-3

4.0

Cl2

4.4. The repulsive parameter n of the LJ potential

C2H6

n-C 5H12

4

C(CH3)4

3

C2 H6 Cl2

2

n-C 4H10

1

CH4 C 3H 8

0 0.0

4.0

4.5

i-C4H10

*

n-C5H12 C(CH3)4

5.5

DR  hRðv ¼ 1Þi
hRðv ¼ 0Þi ¼ 0:89  10
12 m for Cl2 [31]. However, this nearly exact agreement might be accidental. We note that d0 for the hydrocarbons also shows the right order of magnitude compared to values estimated from spectroscopic data. In the case of nepentane, C(CH3 )4 we obtained d0 ¼ 0:7  10
12 m. This value is expected if the enlargement of the molecule is due to the nearly harmonic C–C stretching mode. In the case of the other hydrocarbons d0 is between 1.2 and 2:4  10
12 m. In these cases the change of the molecular size is determined by more anharmonic modes, like torsional vibrations. Another meaningful quantity is the C6 constant of the leading dispersion-interaction energy term
C6 =R
6 . Highly accurate experimental and theoretical values of C6 are available only for the n-alkanes and chlorine [32,33]. In the case of iso-butane and neopentane we have calculated C6 from known polarizability data [24]. The results are compared to C6 ¼ neR6m =ðn
6Þ which result from our present study. In Fig. 12 a nearly linear relationship is observed if the potential parameters of the LJTDP are used. The correlation gets slightly worse if C6 from the LJ parameters is used. We do not expect any numerical agreement between C6 and C6 because the latter contains contributions from all the other
C2n R
2n terms present in the dispersion-interaction energy. It is noteworthy that in the case of SF6 [8] the result for C6 given by Kumar et al. [34] is also in the vicinity of the straight line.

10 C 6 / au (this work)

10

10 Vm

1/3

/m

cules. In order to investigate this effect we try to correlate some of the characteristic potential parameters with other physically meaningful quantities. The equilibrium distance at T ¼ 0 K, Rm ð0Þ, should reflect the van der Waals radius of the molecule. However, this is not a well defined quantity. Instead of this we consider the volume of a molecule derived within the widely used concept of atoms-in-molecules [29]. Within the framework of this theory a molecule is devided into atomic basins. Their surface is defined by a zero-flux in the gradient vector field of the charge density, r.ðrÞ  nðrÞ  0. The resulting volumes of the atomic basins are transferable between molecules of a homologues series, like the alkanes. This P allows for a simple calculation of the volume Vm ¼ i Vm;i of a molecule once the individual contributions are known. Using given values for the volumes Vm;i of the CH3 and CH2 groups [30] the volumes of the alkanes can be readily calculated. The volume is defined by a 0.001 au envelope in the electronic density. In the case of Cl2 we did not find any value of Vm in the literature. Hence a proportional scaling of the volumes and radii of several other molecules are used to deduce Vm of chlorine [29]. In Fig. 11 the cube-root of Vm is plotted against Rm ð0Þ for both the LJTDP and LJ potential. In the case of the LJTDP we observe a very good correlation between these two quantities, which shows that Rm ð0Þ correlates very well with a hypothetic radius obtained from a consistently defined volume of a molecule. On the other hand, the correlation gets worse if Rm ð0Þ of the temperature-independent Lennard-Jones potential is used. The parameter d0 can be interpreted as a mean change of the amplitude Dað0 ! 1Þ of the molecule going from vibrational state v ¼ 0 to state v ¼ 1. We compare the resulting values of d0 to the effective enlargement DR of the anharmonic oscillator. In the case of Cl2 we have obtained d0 ¼ 0:88  10
12 m. The analysis of spectroscopic data yields an enlargement of

201

5.0

10

5.5

10 Rm(0) / m Fig. 11. Cube-root of the molecular volume as a function of Rm ð0Þ. Full squares – LJTDP; open squares – LJ. The dotted line is obtained from a constrained linear Gaussian least-squares fit of the LJTDP solution, which goes through zero.

0.5

1.0

1.5 2.0 -3 10 C6 / au

Fig. 12. Dispersion interaction energy constants C6 obtained in this work as a function of experimental values of C6 . Full squares – LJTDP; open squares – LJ. C6 of iso-butane is identical for both potentials. The dotted line is obtained from a constraint least-squares fit of the LJTDP solution.

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for methane we end up with a value of n  950 for npentane. This means that the corresponding LJ potential is gradually approaching the well-known Sutherland potential [1]. This may explain the failure of the onecenter Lennard-Jones (12 ) 6) and (18 ) 6) potential models used so far for the simultaneous calculation of transport and equilibrium properties of the n-alkanes.

5. Conclusion The isotropic temperature-dependent potential of the Lennard-Jones (n
6) type (LJTDP) is successfully used to approximate simultaneously the measured transport and equilibrium properties of seven pure alkanes, their binary mixtures, and chlorine. The results obtained for non-spherical molecules justify that we have used the same approach as we did before in the case of spherical molecules and their binary mixtures. The parameterised LJTDP could be used to compose tables with reference data for Rm , e, B, b, g, and qD of the pure gases in a wide temperature range. Especially in the case of the unlimited number of binary mixtures of different compositions our approach allows for a fast and reliable calculation of their thermophysical properties. Strong evidence was found for the correlation of the potential parameters of the LJTDP with other meaningful molecular properties. We observe a linear dependence of Rm ð0Þ on the cuberoot of the molecular volume. In our earlier works we have interpreted the mean enlargement d0 of the first vibrational level as a pure fit parameter. In this work we have noticed especially for Cl2 that d0 shows a numerical agreement with the enlargement of the molecule due to vibrational excitation. This observation will be discussed in detail in our future work where the enlargement dðT Þ of the molecule will be calculated from spectroscopic data only. In this case the LJTDP will be reduced to a universal three parameter potential. Further studies should also extend our temperaturedependent potential model to more anisotropic molecules as well as to globular macromolecules.

Acknowledgements Financial support of the Deutsche Forschungsgemeinschaft, the Fonds der Chemischen Industrie and the Bulgarian Academy of Sciences is gratefully acknowledged. Fruitful discussions with Dr. M. Willeke (ETH Zu¨rich) are gratefully acknowledged.

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