ON THE QUESTION OF THE RELATIONSHIP BETWEEN THE ROTATIONAL MOTION OF MOLECULES AND THE ALTERNATION OF PROPERTIES IN HOMOLOGICAL SERIES Текст научной статьи по специальности «Физика»

UDC 544.273

DOI: 10.18698/1812-3368-2022-2-87-101

ON THE QUESTION OF THE RELATIONSHIP BETWEEN THE ROTATIONAL MOTION OF MOLECULES AND THE ALTERNATION OF PROPERTIES IN HOMOLOGICAL SERIES

M.N. Koverda1 E.N. Ofitserov2 S.A. МШаг3 R.V. Yakushin2 V.S. Boldyrev24-5

m.koverda@uniyar.ac.ru

ofitser@mail.ru

sammillar@fieldfare.land

yakushin@muctr.ru

boldyrev.v.s@bmstu.ru

1 P.G. Demidov Yaroslavl State University, Yaroslavl, Russian Federation

2 D. Mendeleev University of Chemical Technology of Russia, Moscow, Russian Federation

3 University of St. Andrews, Scotland

4Bauman Moscow State Technical University, Moscow, Russian Federation 5NPO "Lakokraspokrytie", Khotkovo, Moscow Region, Russian Federation

Abstract

In the paper the emphasis is placed on attempt to describe the effect of alternation of melting points in homologous series using the example of melting points of alkanes of normal structure from the frame of reference of the concept of rotational motion of molecules in a liquid. In the existing literature, the presence of separate sequences for the melting points of even and odd homologues is usually explained by packing effects, in which, due to different symmetry, even and odd representatives crystallize with different densities of the crystal lattice. However, this behavior can also be explained by the difference in the entropies of the rotational motion of molecules in the liquid phase, which is observed due to the different positions of the axis of rotation for even and odd homologues. The theoretically predicted difference in the moments of inertia for molecules with an even and an odd number of carbon atoms in the chain was confirmed by calculating the moments of inertia of n-alkane molecules, for which the atomic coordinates were optimized using density functional theory calculations

Keywords

Even-odd effect, moment of inertia, rotational motion of molecules, entropy of rotational motion, melting point

Received 23.12.2020 Accepted 03.05.2021 © Author(s), 2022

Introduction. The phenomenon of alternation of the absolute values of some physicochemical properties is widely known in homologous series of compounds of the form X (CH2 ) Y, where X and Y are a hydrogen atom, a hydrocarbon radical, or some functional group. The most striking and well-known example is the melting points of substances [1], in which case homologues with an even and odd number n form their own sequences (Fig. 1).

Fig. 1. Alternation of melting points in some homologous series X(CH2 ) Y (1 — X, Y = H; 2 — X = H, Y = CI;

3 — X = H, Y = COOH; 4 — X, Y = COOH)

This phenomenon, also called the "even-odd effect", was first demonstrated by Bayer [2] in the melting points of dicarboxylic acids. As a rule, at present it is customary to explain the even-odd effect on melting of homologues by packing effects [3-7]. It is usually assumed that, due to the different symmetry of the molecules of even and odd homologues, some of them have a more efficient crystal packing, as a result of which the molecular crystals of even or odd homologues have a higher density and, accordingly, a higher melting point. For homologues with a sufficiently large number of carbon atoms in the main chain (above six), Bond has come to the conclusion that this explanation to be quite acceptable [4].

However, it is necessary to take into account that at present, the manifestation of an even-odd effect has been recorded for a sufficiently large number of properties, and both related to the crystalline state of a substance and not. For example, an even-odd effect was found for such properties as spectral characteristics, reaction rate constants, biological properties and solubility, dipole moments, magnetic susceptibility, etc. [8-25].

Thus, at the moment it becomes clear that the even-odd effect is not a separate special case, but is rather widespread in nature. As a rule, the considered alternations of properties always show a single picture: the greatest difference in values is observed at the beginning of the homologous series, and then gradually fades away (see Fig. 1). In this case, the first members of the homologous series most often drop out of the general sequence.

Discussion of the difference between even and odd homologues. In this paper, we attempted to look at the effect of alternation of melting points in homologous series using the example of melting temperatures of normal alkanes from the point of view of the concept of rotational motion of molecules in a liquid. The optimal structure for the solid phase is the one with the minimum total energy. In this case, for temperatures other than absolute zero, it is necessary to take into account the entropy effects. The most stable state of matter will be such that the Gibbs free energy is minimal [26]:

G = U+pV—TS. (1)

With a change in temperature, the difference in the entropies of the phases leads to the fact that one of the phases begins to have a lower free energy and becomes more stable at this temperature, which causes phase transitions between allotropie modifications, as well as melting and crystallization.

Thus, melting can be represented as a competition between two phases, which is schematically shown in Fig. 2. At low temperatures, the solid phase

Fig. 2. Energy change of solid and liquid phases with increasing temperature (1 — solid phase; 2 — liquid phase)

T

1 m

has a lower Gibbs free energy, but at high temperatures the liquid phase has a lower Gibbs free energy due to the entropy term TS in Eq. (1). The phase transition occurs at the point corresponding to the equality of the Gibbs free energies of the two phases, which corresponds to the melting or crystallization temperature of the substance.

For the homologous series of normal alkanes (see Fig. 1), homologues with an odd number of carbon atoms have slightly lower melting points than even ones [27].

The diagram shown in Fig. 2 suggests that from the point of view of the change in the Gibbs free energy, the fundamental difference between even and odd homologues, based on Eq. (1), should be that even or odd homologues should have slightly different values of the entropy S relative to the general trend, while maintaining the constancy of the difference in the even and odd series, due to which the free energy of the liquid phase decreases with increasing temperature at a different rate and the properties of even and odd homologues form their own sequences.

The entropy term in Eq. (1) consists of several components [28]. Since, with a high degree of approximation, we can say that the total energy of a molecule can be represented in the form of mutually independent components

£( — So = £ trans + Srot + ^vib +

then we can divide the molecular partition function into factors

z = . E g,.j,k.i<xP

t,j, к, l

^ £ trans + Erot + ^vib + ^e ^

kT

— Ztrans Zrot 2vib Ze, (2)

where gi,jXl = gtrans £rot £vib ge by virtue °f the assumption of the independence of certain types of motion from each other. For this approximation, the grand partition function a mole of indistinguishable particles is determined by the equation

_ (-Ztrans -Zrot Zvib Ze) _ V --77—;-- trans Vrot Vvib Ve.

AT A!

ZNa

Qtrans Qrot Qvib Qe = —~ T Zro^ Z^j^ Ze A .

A/a '

Since thermodynamic quantities are evaluated in terms of In Q, and In Q is equal to the sum of the logarithms of the partition function of all types of molecular motion, it is possible to calculate the individual contributions of each motion to thermodynamic functions, inch into entropy:

^(T) = ^trans + Srot + Syib + (3)

Which of the entropy terms of Eq. (3) can be the reason for the difference in the properties of even and odd properties of homologues? Let us consider the calculation of each of them for three-dimensional ideal gas molecules. The corresponding values of entropy can be calculated using the components of the molecular partition function from Eq. (2):

Qtrans = к In Qtrans +kT

д In Qt

rans

dT

у

= Я1п ^EUL + rt

Na

^ д In Ztrans ^

дТ

У

;(4)

р

Srot = к In Qrot + кт ^^ = Rln Zrot + RT ; (5)

dT dT

S^b = kin Q^ + к T ^^ = R In Z^b + R T ^^;

On the Question of the Relationship between the Rotational Motion...

Se=k\nQe+kT^^ = RlnZe+RTd]nZe

dT dT

The total partition function of the translational motion is calculated as [31]

00 00 , 00 Ztrans = J e-™2*dix J e~aiydiy J e-^diz =

0 0 0

V(2nmkTf12

— ^ trans, x ^ trans, y trans, z % • V0)

h

Here V" is the volume available for the translational motion of the molecule. As follows from Eq. (6), none of the factors is able to "distinguish" between even and odd molecules. The corresponding value of Strans from Eq. (4) will similarly exhibit a single dependence for both even and odd members of the homologous series.

When calculating the vibrational partition function of a polyatomic molecule, it can be represented as a set of harmonic oscillators, each of which vibrates independently of the others. The number of such oscillators is equal to the number of degrees of freedom / of the molecule per vibrational motion. Then the vibrational partition function of a polyatomic molecule will be

z i=f z i=f 1

^ ,=i ^ ¿=il-exp(-fcvi/(A:r))'

where vt- is the main oscillation frequency of the z'-th oscillator. The value of the vibrational partition function also turns out to be insensitive to the evenness and oddness of homologues.

The electronic partition function also does not distinguish between odd and even molecules. At temperatures T < 103 K, for most atoms and molecules, only the ground unexcited electronic level appears to be populated. Therefore, at room temperature Ze = g0, where go is the statistical weight of the ground electronic level of a molecule or atom.

The most interesting is the consideration of rotational motion. It was shown that the entropy of rotational motion of molecules in the gas phase correlates with some physical properties of the liquid phase, in particular, the volumetric coefficient of thermal expansion [29]. At the same time, attempts to predict the melting points of alkanes on the basis of empirical or thermodynamic models were made earlier [30, 31]. We can draw conclusions that the contribution responsible for the difference between the even and odd homologues from these terms is the entropy of rotational motion, since only it is able to respond to the

different symmetries of even and odd homologues. Let us first consider the calculation of the rotational entropy for a molecule in the gas phase. In this case, the rotational entropy can be calculated from the partition function of the rotational motion (Eq. (5)).

The rotational partition function of a polyatomic nonlinear molecule is a function of the moment of inertia of the molecule, temperature, and also the symmetry of the molecule:

¡8n2IxkT 8n2IykT Sn2IzkT Zl0t=°i h2 h2 h2 ' ()

where Ix, Iy, and Iz represent the moments of inertia of the molecule about the three main axes of rotation. Since

RTd\nZIot = 3 RT = 3R dT 2 T 2 '

then Eq. (5) for calculating Srot will have the form

i

Srot - R

lnZrot+| v 2 У

= -R]n(lxIyIz)+-R\nT-

-Rkia + R

, Vit(8n2kf'2^

— + In -

2 h

\

з

(8)

Equation (8) is strictly valid for molecules in the gas phase. In the liquid phase, unlike gas, the molecules are in constant contact with each other, therefore, due to intermolecular interactions, the real entropy of the rotational motion of the molecules should be lower than that calculated by formula (8). The moment of inertia of molecules about a certain axis, used in Eqs. (7) and (8), can be calculated as the sum of the products of the masses of atoms by the square of their distance to the chosen axis:

I-Em,-»-2. (9)

i

As the x axis, we take the main axis of rotation corresponding to the minimum possible moment of inertia of the molecule Ix. Obviously, in the case of a linear chain, for example, n-alkanes, the x axis will run along the main carbon chain. In this case, even and odd homologues in this case have a fundamental difference, which consists in the fact that they have a different character of the position of this axis (Fig. 3).

Fig. 3. The location of the axis of rotation with the minimum value of the moment of inertia for even and odd carbon chains

VA

-W-

The aim of this research is to find a qualitative correlation between the melting temperatures of n-alkanes and the calculated moments of inertia about the main axis of rotation.

Results and discussion. Even homologues have a symmetric distribution of the heaviest carbon atoms about the x axis, while odd ones are asymmetric, which is why the rotation axis along this direction of the molecule is shifted towards the side with a large number of carbon atoms. Since formula (9) contains the square of the distance to the axis of rotation, this leads to the fact that for odd homologues a proper sequence of moments of inertia should be observed along this axis, which should be located slightly higher than for even ones. Let us show why this is so. Let us consider for simplicity the even and odd chain of carbon atoms and derive equation for them for calculating the moment of inertia about the x axis (Fig. 4).

Fig. 4. Geometrical parameters of carbon chains of even and odd homologues

Let us introduce the following notation: a is the tetrahedral angle, I is the length of the carbon-carbon bond, m is the mass of a carbon atom, b is the distance from a carbon atom to the axis of rotation along the bond line from the side of an even number of carbon atoms in the case of odd hydrocarbons, n is the number of carbon atoms in the chain.

For even chains, the axis of rotation will pass through the middle of the carbon-carbon bond, so it is obvious that

I even — ПУП

1 \2 1, ax

— I cos —

v2 2

For odd chains, the axis of rotation is shifted to the side with a large number of carbon atoms so that the total value of the moment of inertia is minimal. Let us derive an equation for calculating the moment of inertia:

r n~1 ¿odd = —Г— WÎ

( 2 , о п +1 '

cos — b" + -m cos —

1 2 J 2 , 2 )

(l-bf.

(10)

The value of b is found from the condition that the derivative of the moment of inertia with respect to b is equal to zero

a

dlodd ( 1ч ,

-= ( и — 1 ) m cos

db K ' I 2 j

b + (n + l) m

\2

/ \2 ' aл

cos —

ч 2 У

^- = 2nmfcos-] b-(n + l)m/

db I 2 J V '

b-(n + l)ml = 0,

' a42

cos— I =0; v 2

f \2 ' aл

cos — v 2y

from where

2 n

(11)

Then, substituting the value of b from (11) into (10), we obtain

г и-1

hdd =——m

a

cos —

2 j

\2

(n + l)l

\2

2n

n +1 f a

+-ml cos —

2 I 2 у

\2

2n

Having built the graphs of the dependence l-f(n) according to the derived relations for even and odd chains, we observe an even-odd effect decreasing with increasing chain length (Fig. 5).

Fig. 5. Even-odd effect for theoretically

calculated moments of inertia Ix of carbon chains (1 — even; 2 — odd)

Let us ask ourselves how the rotational motion of a molecule changes when it is transferred from the gas phase to the liquid phase. We assumed that, in addition to the appearance of the indicated inter-- molecular interactions, which prevent the n free rotation of molecules, as a result of close contact of molecules with each other, conditions are created, as a result of which the rotation of molecules along certain axes becomes more preferable. The results of work [4], in which X-ray diffraction analysis of homologues of car-boxylic acids of normal structure, located at a temperature close to the melting point, was carried out indicate that, at least near the melting point, homologues

О

with sufficiently long carbon chains are located in a single inhibited linear conformation. Thus, each molecule of such substances can be represented as being in a cell formed by neighboring hydrocarbon molecules (Fig. 6).

Fig. 6. Schematic representation of alkanes in the liquid phase and the cells they occupy

Since alkanes have an elongated molecular structure, presumably in a liquid the most preferable is the rotation of such molecules along the x axis, relative to which the molecule has the least resistance of neighboring molecules and the smallest moment of inertia Ix. Rotational oscillations along the y and z axes are limited by neighboring molecules and the conformational flexibility of the chains, as a result of which rotation of the molecule as a whole is impeded. Most likely, complete rotation does not occur at all [32], but rotational rocking of the molecule occurs relative to the equilibrium position. From formula (8) it is clear that the entropy of rotational motion is directly proportional to the value of the moment of inertia. If, under conditions of close contact of molecules in a liquid, the most probable is rotation (or rotational rocking) about the axis with the smallest moment of inertia, then lower melting temperatures correspond to a larger value of the moment of inertia Ix for odd homologous.

To determine the alternations of the moments of inertia and entropy of the rotational motion of real molecules relative to the x axis, a homologous series of alkanes of normal structure from methane to octane was used. The choice of such model compounds is due to the absence of strong intermolecular (hydrogen bond, Keesom forces, etc.) interactions in the homologous series of alkanes.

At the first stage, optimization of the geometry of molecules was carried out by density functional theory calculation using the PBE method using the 6-311++G(2d,2p) basis set. The calculations were carried out using the GAMESS software package (2019.R1) [33].

At the second stage, the obtained optimized coordinates were used to calculate the moments of inertia in our specially written program "Moments of inertia" [34]. The correctness of the obtained results was verified (for those compounds for which it was possible) by comparison with the database of computational chemistry and comparative tests of the National Institute of Standards and Technology (NIST) [35]. This comparison showed that similar results are obtained using other methods and basis sets.

In the course of the calculations, the following values of the moments of inertia were obtained [36], presented in Table.

Calculated values of the moments of inertia of the rotational motion of molecules in the gas phase for the homologous series of n-alkanes

Compound Moments of inertia, Da-Â2 Moments of inertia, kg • m2 тт, к

h h h h-io-46 ïy- КИ4 Iz ■ ÎO^4

Methane 3.16 0.524 0.00524 0.00524 90.69

Ethane 6.19 25.19 25.19 1.03 0.0418 0.0418 90.30

Propane 17.00 59.93 67.67 2.82 0.0995 0.112 85.46

Butane 21.35 139.39 148.43 3.55 0.231 0.246 134.86

Pentane 29.39 260.87 274.89 4.88 0.433 0.456 143.40

Hexane 34.30 445.71 641.59 5.70 0.74 0.766 177.80

Heptane 41.54 697.46 717.52 6.90 1.16 1.19 182.50

Octane 46.71 1034.09 1056.27 7.76 1.72 1.75 216.30

It turned out that the theoretically predicted alternation of the moments of inertia along the x axis is indeed observed and has a character similar to the typical case of oscillations of properties in homologous series (Fig. 7).

Fig. 7. Calculated alternation of the moments of inertia lx for the homologous series of n-alkanes

For the moments of inertia Iy and Iz, orthogonal to the x axis, the calculated moments of inertia rise very quickly due to the square of the distance to these axes (Fig. 8). However, apparently, these values do not play a role in describing the rotational motion of molecules in the

ir, Da A

0 12345678«

Iy, Da -Ä2 4, Da-Ä2

1000 Г 1000 t

800 J 800 I

600 f 600 J

400 у 400 /

200 У 200 У

j»—-Ж^, A —Ar-""^ l l l l l l

0 12345678л 0 12345678«

a b

Fig. 8. Calculated moments of inertia Iy (a) and Iz (b) for the homologous series

of n-alkanes

liquid phase due to the fact that the rotation of molecules as a whole object along these axes is not realized, and also due to the conformational mobility of the carbon chain.

It should be noted that the nature of the alternation of the moments of inertia of the rotational motion of n-alkanes fully corresponds to the observed pattern of changes in the melting temperatures, when lower melting points correspond to a larger value of the moments of inertia. However, it was possible to show only a qualitative dependence of the melting point on rotational motion, since the magnitude of the decrease in the entropy of rotational motion during the transfer of molecules from the gas phase to the liquid is unknown. In addition, since the moment of inertia of a molecule, strictly speaking, is a dynamic quantity due to conformational transitions and the presence of intermolecular interactions in a liquid, it is necessary to understand that in the case of more complex molecules, a different kind of an even-odd effect can also be observed. When considering such molecules, the following factors must be taken into account:

- the presence of short-range order in a liquid, due to the presence of functional groups and also depending on the general shape and symmetry of molecules;

- the presence of sufficiently strong intermolecular interactions (hydrogen bond, Keesom forces, Debye forces), leading to the formation of associates, which, during rotational rocking, behave differently from the original molecules, since they have a different arrangement of the rotation axes. For example, when dimers of carboxylic acids are formed, the resulting complex has other characteristics of symmetry and can rotate as a whole object.

Conclusion. The damped alternation of many properties in homologous series (as the temperature of melting) can be interpreted on the basis of the concept of rotational motion in a liquid.

There is a qualitative correlation between the melting temperatures of n-alkanes and the calculated moments of inertia relative to the main axis of rotation.

Homologous molecules with a sufficiently long carbon chain in a liquid, due to the presence of closely spaced neighboring molecules and conformational mobility, experience difficulties with rotation about axes located not along the main carbon chain. Because of this, rotational oscillations about the axis with the smallest moment of inertia Ix become the determining rotation for such elongated molecules in a liquid. With respect to this axis, homologues with an even and an odd number of carbon atoms in the chain have different symmetry and exhibit alternation of the moments of inertia, which leads to alternation of the entropy of rotational motion Srot and, consequently, alternation of the melting points.

It is likely that later it will be possible to explain other properties that alternate in homological series.

Acknowledgements

The authors express their deep gratitude to P.S. Zhukovfor the computing resources provided and to E. V. Mironovafor help in the selection of materials for the work.

REFERENCES

[1] Haynes W.M., Lide D.R., Bruno T.J. CRC handbook of chemistry and physics. CRC Press, 2017.

[2] Baeyer A. Ueber Regelmassigkeiten im Schmelzpunkt homologer Verbindungen. Chem. Ber., 1877, vol. 10, no. 2, pp. 1286-1288.

DOI: https://doi.org/10.1002/cber.18770100204

[3] Boese R., Weiss H.C., Blaser D. The melting point alternation in the short-chain n-alkanes: single-crystal X-ray analyses of propane at 30 K and of «-butane to n-nonane at 90 K. Angew. Chem. Int. Ed. Engl, 1999, vol. 38, iss. 7, pp. 988-992. DOI: https:// doi.org/10.1002/(SICI)1521-3773(19990401)38:7<988::AID-ANIE988>3.0.CO;2-0

[4] Bond A.D. On the crystal structures and melting point alternation of the n-alkyl car-boxylic acids. New J. Chem., 2004, vol. 28, iss. 1, pp. 104-114.

DOI: http://dx.doi.org/10.1039/B307208H

[5] Bond A.D. Inversion of the melting point alternation in n-alkyl carboxylic acids by co-crystallization with pyrazine. CrystEngComm, 2006, vol. 8, iss. 4, pp. 333-337. DOI: https://doi.org/10.1039/B517513E

[6] Yang K., Cai Z., Jaiswal A., et al. Dynamic odd-even effect in liquid n-alkanes near their melting points. Angew. Chem., 2016, vol. 55, iss. 45, pp. 14090-14095.

DOI: https://doi.org/10.1002/anie.201607316

[7] von Sydow E., Ekwall P., Larsen I., et al. Crystallization of normal fatty acids. Acta Chem. Scand., 1955, vol. 9, pp. 1685-1688.

DOI: https://doi.org/10.3891/acta.chem.scand.09-1685

[8] Lermo E.R., Langeveld-Voss B., lanssen R., et al. Odd-even effect in optically active poly(3,4-dialkoxythiophene). Chem. Commun., 1999, no. 9, pp. 791-792.

DOI: https://doi.org/10.1039/a901960j

[9] Costa J., Fulem M., Schroder B., et al. Evidence of an odd-even effect on the thermodynamic parameters of odd fluorotelomer alcohols. /. Chem. Thermodyn., 2012, vol. 54, pp. 171-178. DOI: https://doi.Org/10.1016/j.jct.2012.03.027

[10] Mishra M.K., Varughese S., Ramamurty U., et al. Odd-even effect in the elastic modulii of a,G)-alkanedicarboxylic acids. J. Am. Chem. Soc., 2013, vol. 135, iss. 22, pp. 8121-8124. DOI: https://doi.org/10.1021/ja402290h

[11] Asai O., Kishita M., Kubo M. Magnetic susceptibilities of the cupric salts of some a,co-dicarboxylic acids. /. Phys. Chem., 1959, vol. 63, iss. 1, pp. 96-99.

DOI: https://doi.org/10.1021/jl50571a024

[12] Geiseler G., Pilz E. Zur Kenntnis der Dipolmomente homologer a-Olefine. Chem. Ber., 1962, vol. 95, iss. 1, pp. 96-101. DOI: https://doi.org/10.1002/cber.19620950118

[13] Gopalakrishna C., Haranadh C., Murty C. Dipole moments of some cholesteryl compounds. /. Chem. Soc. Faraday Trans., 1967, vol. 63, pp. 1953-1958.

DOI: https://doi.org/10.1039/tf9676301953

[14] Anderson H., Hammick D. Interaction between polynitro-compounds and aromatic hydrocarbons and bases. Part X. Picric acid and alkylated benzene derivatives in chloroform solution. J. Chem. Soc., 1950, pp. 1089-1094.

DOI: https://doi.org/10.1039/jr9500001089

[15] Cardwell H., Kilner A. Elimination reactions. Part I. Acid-catalysed enolisation and substitution reactions of ketones. J. Chem. Soc., 1951, pp. 2430-2441.

DOI: https://doi.org/10.1039/JR9510002430

[16] Gero A. Studies on enol titration. II. Enol contents of some ketones and esters in the presence of methanol. /. Org. Chem., 1954, vol. 19, iss. 12, pp. 1960-1970.

DOI: https://doi.org/10.1021/jo01377a013

[17] Yamana T., Mizukami Y., Tsuji A., et al. Studies on the stability of amides. I. Hydrolysis mechanism of N-substituted aliphatic amides. Chem. Pharm. Bull., 1972, vol. 20, iss. 5, pp. 881-891. DOI: https://doi.org/10.1248/cpb.20.881

[18] Okamoto K., Kita T., Shingu H. Kinetic studies of bimolecular nucleophilic substitution. III. Rates of Sn2 and E2 reactions of co-chloroalkylbenzenes with sodium acetate in 50 vol% aqueous acetone — alternation effects in saturated carbon chains. Bull. Chem. Soc. Jpn., 1967, vol. 40, no. 8, pp. 1908-1912. DOI: https://doi.org/10.1246/bcsj.40.1908

[19] Kumar N., Chaudhary S., Upadhyay P., et al. Even-odd effect of the homologous series of nCHBT liquid crystal molecules under the influence of an electric field: a theoretical approach. Pramana — J. Phys., 2020, vol. 94, art. 106.

DOI: https://doi.org/10.1007/sl2043-020-01967-0

[20] Ojha D.P., Kumar D., Pisipati V. Odd-even effect in a homologous series of 4-n-alkylbenzoic acids: role of anisotropic pair potential. Z. Naturforsch. A, 2002, vol. 57, iss. 3-4, pp. 189-193. DOI: https://doi.org/10.1515/zna-2002-3-411

[21] Pines A., Ruben D.J., Allison S. Molecular ordering and even-odd effect in a homologous series of nematic liquid crystals. Phys. Rev. Lett., 1974, vol. 33, iss. 17, pp. 1002-1005. DOI: https://doi.org/10.1103/PhysRevLett.33.1002

[22] Yamamoto R., Ishihara S., Hayakawa S., et al. The even-odd effect in the Kerr effect for nematic homologous series. Phys. Lett. A, 1977, vol. 60, iss. 5, pp. 414-416.

DOI: https://doi.org/10.1016/0375-9601(77)90037-8

[23] Chang R., Jones F.B., Ratto J.J. Molecular order and an odd-even effect in an homologous series of schiff-base nematic liquid crystals. Mol. Cryst. Liq. Cryst., 1976, vol. 33, iss. 1-2, pp. 13-18. DOI: https://doi.org/10.1080/15421407608083866

[24] Pradeilles J.A., Zhong S., Baglyas M., et al. Odd-even alternations in helical propensity of a homologous series of hydrocarbons. Nat. Chem. 2020, vol. 12, pp. 475-480. DOI: https://doi.org/10.1038/s41557-020-0429-0

[25] Lee S.K., Heo S., Lee J.G., et al. Odd-even behavior of ferroelectricity and antiferro-electricity in two homologous series of bent-core mesogens. /. Am. Chem. Soc., 2005, vol. 127, iss. 31, pp. 11085-11091. DOI: https://doi.org/10.1021/ja052315q

[26] Hofmann P. Solid state physics. An introduction. Wiley, 2015.

[27] Le Neindre B., Garrabos Y. Scaled equations for the coexistence curve, the capillary constant and the surface tension of n-alkanes. Fluid Phase Equilib., 2002, vol. 198, iss. 2, pp. 165-183. DOI: https://doi.org/10.1016/S0378-3812(01)00737-3

[28] Chandler D. Introduction to modern statistical mechanics. Oxford Univ. Press, 1987.

[29] Koverda M., Ofitserov E., Shilanova E., et al. Using the molecular rotational motion concept to predict the volume expansion coefficients of liquids. Phys. Chem. Liquid., 2021, vol. 59, iss. 5, pp. 770-780. DOI: https://doi.org/10.1080/00319104.2020.1814774

[30] Chickos J.S., Nichols G. Simple relationships for the estimation of melting temperatures of homologous series. J. Chem. Eng. Data, 2001, vol. 46, iss. 3, pp. 562-573.

DOI: https://doi.org/10.1021/je0002235

[31] Flory P.J., Vrij A. Melting points of linear-chain homologs. The normal paraffin hydrocarbons. /. Am. Chem. Soc., 1963, vol. 85, iss. 22, pp. 3548-3553.

DOI: https://doi.org/10.1021/ja00905a004

[32] Frenkel J. Kinetic Theory of Liquids. Peter Smith Publisher, Incorporated, 1984.

[33] Barca G., Bertoni C., Carrington L. Recent developments in the general atomic and molecular electronic structure system. /. Chem. Phys., 2020, vol. 152, iss. 15, art. 154102. DOI: https://doi.Org/10.1063/5.0005188

[34] Koverda M. Program Moment of inertia. Mendeley Data, 2020, Version 1. DOI: https://doi.Org/10.17632/skj4999wr6.l

[35] Russell D. NIST computational chemistry comparison and benchmark database. NIST Standard Reference Database Number 101. Release 21.

DOI: https://doi.org/10.18434/T47C7Z (accessed: 15.01.2022).

[36] Koverda M. On the question of the relationship between the rotational motion of molecules and the alternation of properties in homological series. Additional materials for the article. Mendeley Data, 2020, Version 1.

DOI: https://doi.org/10.17632/3n6wr3kkbx. 1

Koverda M.N. — Post-Graduate Student, Department of Organic Chemistry, Faculty of Biology and Ecology, P.G. Demidov Yaroslavl State University (Sovetskaya ul. 14, Yaroslavl, 150003 Russian Federation).

Ofitserov E.N. — Dr. Sc. (Chem.), Professor, Department of Chemistry and Technology of Biomedical Preparations, Faculty of Chemical Pharmaceutical Technologies and Biomedical Preparations, D. Mendeleev University of Chemical Technology of Russia (Miusskaya ploshchad 9, Moscow, 125047 Russian Federation).

Millar S.A. — Graduate, School of Biology, University of St. Andrews (College Gate, St. Andrews, KY16 9AJ Scotland).

Yakushin R.V. — Cand. Sc. (Eng.), Assoc. Professor, Department of Chemistry and Technology of Biomedical Preparations, Faculty of Chemical Pharmaceutical Technologies and Biomedical Preparations, D. Mendeleev University of Chemical Technology of Russia (Miusskaya ploshchad 9, Moscow, 125047 Russian Federation).

Boldyrev V.S. — Cand. Sc. (Eng.), Assoc. Professor, Department of Chemistry, Bau-man Moscow State Technical University; Head of the Department, Engineering of Chemical and Technological Systems, Engineering Center Automation and Robotics, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5, str. 1, Moscow, 105005 Russian Federation); Master Student, Faculty of Digital Technology and Chemical Engineering, D. Mendeleev University Technology of Russia (Miusskaya ploshchad 9, Moscow, 125047 Russian Federation); Director's Adviser, Research Institute, NPO "Lakokraspokrytie" (Khudozhestvennyy proezd 2e, Moscow Region, Khotkovo, 141370 Russian Federation).

Please cite this article as:

Koverda M.N., Ofitserov E.N., Millar S.A., et al. On the question of the relationship between the rotational motion of molecules and the alternation of properties in homological series. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2022, no. 2 (101), pp. 87-101. DOI: https://doi.org/10.18698/1812-3368-2022-2-87-101