Structural analysis of liquid 1,4-dimethylbenzene at 293 K

Journal of Molecular Structure 783 (2006) 204–209 www.elsevier.com/locate/molstruc

Structural analysis of liquid 1,4-dimethylbenzene at 293 K H. Drozdowski Optics Laboratory, Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan´, Poland Received 31 May 2005; accepted 8 August 2005 Available online 17 October 2005

Abstract Structural analysis of liquid 1,4-dimethylbenzene C6H4(CH3)2 by X-ray monochromatic radiation scattering method was performed. The X-ray measurements were made at room temperature for the scattering angle range Q varying from 3 to 608. The most probable parameters of 1,4dimethylbenzene molecules were determined from the detailed analysis of the distributions of scattered X-radiation intensity. Experimental n P function iðSÞZ ½Ieu ðSÞKN nj fj2 ðSÞ=Nfe2 ðSÞ was determined by X-ray scattering from a liquid sample and compared with the theoretical function j

im(S) calculated for the most probable assumed model of the 1,4-dimethylbenzene structure. The theoretical and experimental curves were in good n P agreement, thus confirming the proposed structure of this molecule. The radial distribution function of electron density 4pr 2 K j ½rk ðrÞKr0  was j;k

calculated according to Warren and some intra- and intermolecular distances in liquid 1,4-dimethylbenzene were determined. From the position of the main maximum in the scattered radiation intensity distribution the mean of the least intermolecular distance R was calculated, to find the volume V of the unit cell of liquid 1,4-dimethylbenzene. The structural data obtainable by X-ray analysis for the liquid studied were discussed. q 2005 Elsevier B.V. All rights reserved. PACS: 61.25.Em Keywords: Molecular structure function; Reduction method of Blum and Narten; Electron-density radial-distribution function

1. Introduction The paper reports the structural study of 1,4-dimethylbenzene C6H4(CH3)2 (melting point 286 K, boiling point 411 K) at a temperature of 293 K. The most probable model of the structure of short-range order was obtained using the method of exact analysis of the distribution curve in terms of functions of reduced intensity. The application of the reduction method has been described by Blum and Narten [1]. The earlier reported structural data for liquid 1,4dimethylbenzene have prompted the study of intermolecular interactions in diluted binary mixtures of benzene nitro derivatives: o-nitroanisole and o-anisidine in 1,4-dimethylbenzene or in benzene [2]. Biswas [3] reported the elementary cell parameters of the ˚ , bZ8.45 A ˚ , cZ 1,4-dimethylbenzene crystal aZ7.56 A 0 ˚ 11.11 A, bZ98857 . The compound crystallizes in the 2 monoclinic system in the space group C2h ðP21 =mÞ, and contains four molecules in the elementary cell.

The structure of liquid 1,4-dimethylbenzene was studied for the first time using Mo Ka radiation in the range 3.08%Q%608, performing the analysis of radial functions in ˚. the range 0! r % 20 A X-ray diffraction study of liquids is based on the Fourier analysis of the reduced intensity I(S) function, defined [4] as: iðSÞ Z im ðSÞ C id ðSÞ;

(1)

where im(S) is the molecular structure function describing the scattering by a single molecule and id(S) is the distinct structure function providing the information about intermolecular correlations from the experimental data. The reduced intensity i(S) represents the structurally sensitive part of the total  coherent intensity IðSÞin electron units. The values of the structural and physical parameters of liquid 1,4-dimethylbenzene are collected in Table 1. Dimethylbenzene samples of 99% purity were purchased from Aldrich– Chemie (Germany). 2. Experimental

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X-ray scattering in liquid 1,4-dimethylbenzene was ˚. measured by applying Mo Ka radiation, lZ0.71069 A Monochromatisation was obtained with a graphite crystal.

H. Drozdowski / Journal of Molecular Structure 783 (2006) 204–209

205

Table 1 Physical and structural parameters of liquid 1,4-dimethylbenzene P ( j Zj denotes the sum of the atom numbers in one molecule) Mean effective number of electrons per hydrogen atom K H Mean effective number of electrons per carbon atom K C P Total effective number of electrons in one molecule j K j P Total of atom numbers in one molecule j Zj Macroscopic density (g/cm3) Molecular mass (g/mol) ˚ 3) Mean electron density (el/A

0.431 6.734 58.000 58 0.866 106.17 0.285

The scattered intensity distribution was measured by a goniometer HZG-3 for the angles 38%Q%608 at every 0.28, where 2Q is the scattering angle. The number of pulses in a chosen period of time was calculated and, thus obtained pulse density was recorded. Detail conditions of the X-ray diffractometer work are described in Table 2. The measurements were performed using the transmission technique with the incident and diffracted beams symmetric upon the flat sample surface [5]. The geometry of the X-ray path is shown in Fig. 1(a). The dimethylbenzene under investigation was placed in a flat double thermostated cuvette (Fig. 1(b)) closed by mica windows of thickness 0.025G 0.001 mm. The distance between the mica windows was 1.00G0.01 mm and the temperature was maintained at 293.0G0.1 K. Good X-ray diffraction patterns were obtained at anodic voltage 55 kV on the valve and anodic current 20 mA (Table 2). The X-ray lamp focal point, the sample studied and the input slit of the counter lie on the common focusing Rowland circle. The movement of the counter on the goniometer circle is coupled with the rotation of the table with the sample about the vertical axis of the goniometer at the ratio 2:1. 3. Correcting the experimental data The experimentally obtained function of the angular distribution of the scattered X-ray intensity was corrected to include the background [6], polarisation [7], absorption [8] and anomalous dispersion [9]. Table 2 Conditions of scattered radiation intensity measurements on an X-ray diffractometer Lamp Voltage Anodic current intensity Power Range of angles measured Range of measurements in SZ4p(sin Q/l) Distances between the measuring points Time of pulse counting Rate of pulse counting Slit (divergence, counter) Graphite monochromator Monochromatisation angle Qm

˚) Mo (lZ0.71069 A 55 kV 25 mA 1300 W 3–608 ˚ K1 (0.430–15.310) A 0.28 40 s 3!105/s 28; 28 6800 0

Fig. 1. (a) A scheme of the goniometer measuring system in the transmission method: (1) X-ray lamp anode; (2) monochromator; (3) the set of slits restricting the original beam; (4) cell with the liquid; (5) the set of input slits of the counter; (6) Soller slits; (7) proportional counter; (8) measuring wheel of the goniometer. (b) The doubly thermostated measuring cuvette: (1) brass block; (2) thermostating blocks; (3) the measuring cell with a preparation.

For a flat preparation of the thickness D, the absorption coefficient applied in the transmission method was [10]: Að2QÞ Z

mDðsec 2QK1Þ ; 1Kexp½KmDðsec 2QK1Þ

(2)

where m is the linear absorption coefficient of a liquid studied, D is the thickness of the sample studied and 2Q is the scattering angle. The original beam of X-ray radiation was monochromatised by reflection from a planar graphite crystal and the polarisation factor was calculated from the equation [11]: Pð2QÞ Z

1 C cos2 2Qm cos2 2Q ; 1 C cos2 2Qm

(3)

where 2Q is the angle at which the X-ray beam was reflected from the monochromator surface. In our experiment when the Mo Ka X-ray beam was reflected from the plane (002), the angle was QmZ6800 0 . The scattered X-radiation was normalised to electron units [e.u.] according to the Krogh-Moe [12] and Norman [13] method. The fundamental idea of normalisation is based on the finding that for large scattering angles all interference—both interatomic and intermolecular—disappears. Then, for the angular range considered, the intensity distribution of the experiment covers the theoretical curve. The calculation of the theoretical curve requires knowledge of the atomic

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H. Drozdowski / Journal of Molecular Structure 783 (2006) 204–209

composition of the smallest structural unit, the dimethylbenzene molecule. For a given atomic composition of a scattering structural unit, the curve of intensity of independent scattering can be obtained from tabulated atomic scattering factors [14]. The shape of this curve depends on a given scattering structural unit. For the liquid studied the following normalisation relationship was obtained: N N ð n ð X C IðSÞS2 dS Z fj2 ðSÞS2 dS; jZ1

0

(4)

0

in which C is a normalisation coefficient and fj(S) stands for atomic scattering factors expressed in electron units [15].

4. Calculations The X-ray structural analysis of liquids is based on finding the function of radial distribution of electron density. It is found from the following equation with the experimentally determined intensity of X-ray radiation scattered in the liquid [16]: 4pr

n X 2

K j rk ðrÞ Z 4pr

n X 2 jZ1

j;k

2r K j r0 C p

N ð

SiðSÞ sinðSrÞdS; 0

(5) where r is the distance from an atom or molecule selected as scattering centre, rk(r) the function of radial electron density. ˚ 3) is given The mean number of electrons in a unit volume (1 A by the formula [17]: P 10K24 dNA j Zj ; (6) r0 Z M where d is the macroscopic density of the liquid, NA, the Avogadro constant, Zj, denotes the number of electrons in the jth atom and M the molar mass (Table 1). In Eq. (5), the function I(S) is defined [18] as: iðSÞ Z ½Ieu ðSÞKN

n X

nj fj2 ðSÞ=Nfe2 ðSÞ;

(7)

j

where the function Ieu ðSÞrepresents the experimental averaged values of the angular distribution of scattered intensity in electron units, nj is the number of jth type atoms in the molecule, fj, the atomic scattering factor, and fe, the mean scattering factor per electron, N, the number of molecules. The sum of squares of atomic scattering factors is determined by the expression: " #2 " #K2 X X 2 fe ðSÞ Z nj f ðSÞ nj Z : (8) j

j

The effective mean atomic numbers K j appearing in Eq. (5) for the radial distribution and ultimately applied in the calculations are S functions defined in the integration limits (S1, S2) by

the mean values obtained from the expression: K j Z

1 S2 KS1

Sð2

Kj dS;

(9)

S1

where SZ4psin Q/l, l the X-ray scattering wavelength and Q the Bragg angle. The calculations were performed for a finite ˚ K1 toS2Z14.311 A ˚ K1. The range of S values from S1Z0.430 A numerical calculations of the radial electron density distribution function rk(r) were performed applying Simpson’s ˚ analytical method in computing the integrals for 0! r % 20 A ˚ varying by steps of 0.05 A. In Eq. (5) on the right side, the first term describes the electron density radial distribution function for independent scattering in a gas medium, the second term describes the diffraction scattering dependent on the degree of ordering of molecules in a given liquid. The solution of Eq. (5) is thus a function oscillating about the mean electron density and the maxima of this function correspond to the intra- and intermolecular interatomic distances. The least mean intermolecular distance in the liquid can be found from the following expression [19]: 7:73 R Z K0:3; Smax

(10)

where SmaxZ4psin Qmax/l, 2Qmax is the scattering angle determining the position of the main maximum of the scattered ˚ radiation intensity, and l is the X-ray wave length of 0.71069 A in our experiment. On the basis of the mean value of the volume of the first coordination sphere V and the specific volume of the molecules V0, the packing coefficient k of molecules in the liquid is calculated from the formula [20]: V (11) k Z N 0 ; V where N is the mean number of molecules in the volume of the  The specific volume of the molecules first coordination sphere V. V0 was determined on the basis of the volume increments corresponding to the contributions of particular atoms in the total volume. The mean volume per one molecule of the liquid is calculated from the known macroscopic density, molecular mass and the Avogadro number according to the formula [21]: V max Z

M ; NA d

(12)

where M is the molecular mass of the liquid (g/mol), d the liquid density at a certain temperature and NA, the Avogadro constant (molK1). The volume Vmax is the maximum because the liquid density d in formula (12) concerns the whole volume of the liquid, including microvoids characteristic of the liquid phase [22]. 5. Results  of the liquid The normalised angular distribution function IðSÞ studied (Fig. 2a) has one main and some smaller diffuse maxima.

H. Drozdowski / Journal of Molecular Structure 783 (2006) 204–209

207

Fig. 2. (a) Normalised, experimental curve of angular distribution of X-ray scattered intensity. (b) Total independent scattering curve.

˚ K1, which The value of the main maximum is SmaxZ1.22 A  5:15 A ˚, corresponds to the mean shortest interatomic distance dZ obtained from the Bragg equation [17]. The theoretical curve (Fig. 2b) was obtained as a sum of the intensities n n P P Icoh ðSÞZ fj2 ðSÞ of coherent scattering and Iincoh ðSÞZ Ijincoh jZ1

jZ1

ðSÞ of incoherent scattering [11]. The least mean intermolecular ˚ . Small-angle  6:04 A distance calculated from Eq. (10) is RZ scattering result (08!Q%38) was extrapolated to the origin of the coordinate system using second-order functions.  values (Fig. 2) the values On the basis of the experimental IðSÞ of i(S) or the total structural function were calculated from Eq. (7). The molecular structural function im(S) was calculated from the equation proposed by Debye [23] for the Bragg angle Q changing in the range 08–608. Having determined i(S) and im(S), the

Fig. 4. A model of 1,4-dimethylbenzene C6H4(CH3)2 molecule structure.

function id(S) was obtained from Eq. (1). The dependencies of i(S), im(S) and id(S) on S are presented in Fig. 3. The values of im(S) shown in Fig. 3 were calculated from the Debye equation assuming the rigid model of the 1,4dimethylbenzene molecule. This model is shown in Fig. 4, and Table 3 gives the values of the Debye equation parameters: 

im ðSÞ Z 2fCH3 C 4fCH C 2fC

" n X n K2 X uc

isj

# sinðSr ij Þ fi fj Aij : Srij (13)

In this equation, rij is the distance between two atoms i and j (which may or may not be linked by a chemical bond), and Aij is an exponential term which allows for the fact that the atoms within the n-atomic molecule are not strictly at rest but are Table 3 The values of parameters of 1,4-dimethylbenzene molecule model applied in Debye formula Eq. (13). Atom notations the same as in Fig. 4 ðuij Þ denotes the root-mean-square variation in the distance between pairs of atoms [24]) No.

Type of intramolecular interactions

Intramolecular ˚) distances R ij (A

Mean amplitudes ˚) uij (A

1

C1–C2, C1–C6, C2–C3, C3–C4, C4–C5, C5–C6 C1–C7, C4–C8 C1/C3, C1/C5, C2/ C4, C2/C6, C3/C5, C4/C6 C2/C7, C6/C7, C3/ C8, C5/C8 C1/C4, C2/C5, C3/C6 C2/C8, C6/C8, C3/ C7, C5/C7 C1/C8, C4/C7 C7/C8

1.40

0.034

1.54 2.40

0.037 0.057

2.55

0.060

2.79 3.85

0.066 0.089

4.43 5.87

0.101 0.134

2 3

4 5 6 Fig. 3. Curve A (continuous line), the experimental structure function Si(S). Curve B (broken line), the molecular structure function Sim(S) calculated according to Debye. Curve C (dotted line), subtraction of the calculated curve B from the curve A.

7 8

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H. Drozdowski / Journal of Molecular Structure 783 (2006) 204–209

Fig. 5. The electron-density radial-distribution function 4pr2 r0  for liquid 1,4-dimethylbenzene.

P

j;k

K j ½rk ðrÞK

vibrating with respect to each other. Aij has the form exp½Kuij =2S2 , where u ij is the mean-square variation in the distance rij between pairs of atoms. In Eq. (13) fi and fj are the atomic scattering factors for the ith and jth atoms. For calculations of mean amplitudes of vibrations u ij of different pairs ofatoms of a liquid studied, the empirical formula of Mastryukov and Cyvin was applied [24] u ij Z a C br C cr 2 ;

(14)

where r is the internuclear distance in the molecule, and a, b, c are constants equal aZ0.0013837; bZ0.023398; cZK 0.000147. This formula was proposed by Mastryukov andCyvin on the basis of a large body of data from electron diffraction studies. P The plot of 4pr 2 j;k K j ½rk ðrÞKr0  is presented in Fig. 5. The maxima in the radial distribution function correspond to the intra- and intermolecular distances for the liquid 1,4dimethylbenzene. 6. Discussion Analysis of the plots of the experimental i(S) and the calculated im(S) for the model proposed (Table 3, Fig. 4) shows ˚ K1 (Fig. 3). This that these two curves coincide for SR5 A coincidence proves that the proposed model of the 1,4-dimethylbenzene molecule and hence also the interatomic distances obtained, are correct. The shape of the 1,4-dimethylbenzene molecule is described by the values of the three radii defined in the orthogonal projections of the molecule onto three mutually perpendicular planes [25]. Such model can be constructed on the basis of the bond lengths within the molecule and the van der Waals radii of C and H atoms. The methyl groups have the C3V symmetry and lie along the C1/C4 axis (Fig. 4). The assignment of intramolecular distances to the maxima (Fig. 5), has been performed assuming that the methyl group does not have a

significant influence on the rigidity and interatomic distances in the benzene ring. This assumption follows from the results of the theoretical study on the ring deformations under the effect of monosubstitutions. The quantum-chemical calculations performed by Scharfenberg [26] for 22 monosubstituted benzene derivatives have shown that the greatest changes ˚. between the carbon atoms in the ring are of an order of 0.01 A A molecule of 1,4-dimethylbenzene takes on average a volume ˚ 3, whereas an increment of the molecule VmaxZ203.6 A ˚ 3. The packing coefficient, volume [21] is VincrZ124.1 A defined by Kitaigorodsky [21] as the ratio of the specific volume of the molecule to the volume per a molecule in a given liquid, takes a value of 0.60. The value falls within the range determined for the liquid phase—from 0.51 to 0.68. The radial distribution function for 1,4-dimethylbenzene ˚ . In the range (Fig. 5) reveals six maxima in the range up to 20 A ˚ of the argument below 1 A, the values of the radial distribution function making the interpretation difficult or even impossible (for example the negative ones) are not uncommon. They seem to be due to some approximations assumed in the method, e.g. to the extrapolation of small angle scattering results [27]. ˚ , are interpreted The maxima in the range of 3:50% r ! 6:00 A as due to intra- and intermolecular diffraction, whereas those ˚ are due to intermolecular diffraction. for r R 6:00 A The four maxima of this function at 1.10, 1.55, 2.55 and ˚ , are assigned to the distances between the following 4.50 A atom pairs: C–H, C1–H7, C1/C7 and C1/C8, in the molecule ˚ brings studied (Table 3). The maximum at r Z 11:05 A information about the difference between the observed and the average distribution of electron density. ˚  6:04 A The radius of the first coordination sphere RZ determined from the intensity distribution function corre˚ . The maximum at sponds to the fifth maximum at r Z 6:05 A ˚ should be interpreted as a result of molecular r Z 6:05 A diffraction, because the least mean intermolecular distance ˚ . This result is in agreement  6:04 A according to Eq. (10) is RZ with the position of the maximum on the electron-density radial-distribution function (Fig. 5), which confirms the correctness of the electron density radial distribution shown in Fig. 5 and obtained for liquid 1,4-dimethylbenzene. Taking into regard the molecule size (Fig. 3) this results suggests that in liquid 1,4-dimethylbenzene the neighbouring molecules assume the conformation in which the planes of their benzene rings are parallel. The mean least inter- and intramolecular distances were determined with the following accuracy: for ˚ : Dr ZG0:05 A ˚ , for 2! r % 3 A ˚ : Dr ZG0:10 A ˚ , for 1! r % 2 A ˚ : Dr Z 0:12 A ˚ [28]. The random error of the radial r O 3 A distribution function does not exceed 3%. 7. Conclusions This paper is devoted to determination of a structural model of 1,4-dimethylbenzene using the reduction method, which permits a more exact interpretation of the X-ray diffraction data. The method of X-ray scattering and Fourier analysis enabled determination of the mean structural parameters (the intra- and intermolecular distances, the radius of the first coordination

H. Drozdowski / Journal of Molecular Structure 783 (2006) 204–209

sphere, the packing coefficient) of the liquid studied. The use of short-wave radiation from an X-ray tube with a molybdenum anode permitted determination of the sphere of intermolecular ordering. From the known unit cell volume V as well as the specific volume of the molecule V0 and taking into regard the structural model of intermolecular interactions the most probable value of the packing coefficient of the molecules was found to be  0:60. This value falls within the range of k values acceptable kZ for the liquid phase substances. The appearance of clear maxima on the angular and radial functions indicates thepresence of short-range ordering in ˚. liquid 1,4-dimethylbenzene up to the distance of about 20 A The three maxima corresponding to the C–C distances appear ˚ for the molecule studied. The at 1.55, 2.55 and 4.50 A maximum assigned to the C–H bond corresponds to the ˚ within the ring. The values of r Z 6:05 and distance 1.10 A ˚ r Z 11:05 A determined provide the information about intermolecular spatial configurations in the liquid studied. The mean intermolecular distances determined by the reduction method are in good agreement with the maxima on the electron-density radial-distribution function. The least mean intermolecular distance was found from the Voigtlaen˚. der–Tetzner formula [19] as6.04A The structure of liquid 1,4-dimethylbenzene is determined by the presence of the benzene ring. We suggest that in liquid 1, 4-dimethylbenzene at 293 K, the neighbouring molecules assume the configuration with their benzene ring planes in parallel to one another. The parallel conformation is probably assumed because it is more energetically favourable and enables a more favourable packing of the molecules. These results are also consistent with the values presumed in the conformational structure investigation of 1,4-dimethylbenzene performed by empirical and semi-empirical MO-LCAO calculations [29]. The results have also been confirmed by the conformation analysed based on minimisation of the potential energy [30]. Because of the supposed role of the benzene ring in mutual configurations of molecules in liquid 1,4-dimethylbenzene, it seems very probable that the proposed model of local arrangement can also hold for other derivatives of benzene in the liquid phase.

209

Acknowledgements The author wishes to thank Mr M. Kro´lik for his assistance in measurements and performing numerical computations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

[28] [29] [30]

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