^{1}

^{1}

^{*}

^{1}

Basing on the DFT calculations we propose the new theoretical model which describes both the surface tension σ of the short chain n-alkanes at their normal boiling points and their reaction rate constants with hydroxyl radicals OH
• (at 297 ± 2 K) on the basis of their molecular orbital electronic characteristics. It has been shown that intermolecular dispersion attraction within the surface liquid monolayer of these compounds, as well as their reaction rate constants k with OH
• radicals are determined by the energies
*E _{orb}* of the specific occupied molecular orbitals which are the same in the determination of both the above physico-chemical characteristics of the studied n-alkanes. The received regression equations confirm the theoretically found dependences between the quantities of σ and k and the module |

*E*|. For the compounds under study this fact indicates the key role of their electronic structure particularities in determination of both the physical (surface tension) and the chemical (reaction rate constants) properties.

_{orb}As is known, many physicochemical properties of chemical compounds to a large extent related to some characteristic features of their electronic structure, including, first of all, certain occupied frontier Molecular Orbitals (MOs). On the one hand, these MOs can determine such physical properties of liquid chemical compounds as surface tension [

In particular, in our previous work [

The frontier molecular orbitals can also play an important role in the determination of the reaction rate constants for the interaction of structurally related compounds with a common reagent. So, for instance, it in the work [^{●}. It can be assumed that in the case of the interaction “hydrocarbon-OH radical” there is also a certain occupied molecular orbital, which is responsible for this process.

In this connection the main goals of the present work are twofold:

1) to understand the mechanism of formation of monomolecular surface layers for the liquids under consideration and to find an equation connecting the surface tension of n-alkanes at their normal boiling points with calculated molecular electronic structure parameters;

2) to clarify the transition state for the interaction “hydrocarbon-OH radical” and to determine the specific molecular orbitals of n-alkanes which play the crucial role in this reaction. Then, based on this knowledge, to establish a quantitative relationship between properties of these specific MOs and the reaction rate constants for the above interaction.

In our previous work [

σ = χ | E o r b | N V − 2 / 3 ( 1 − T / T c ) , (1)

where V is the molar volume of the liquid taken at a certain absolute temperature T, close to or equal to its normal boiling point, T_{c} is its critical temperature, and N is the total number of molecular centers which form one-particle πσ*- hyperconjugated electronic system. The quantity |E_{orb}| in Equation (1) is the absolute value of the energy of the molecular orbital (MO) responsible for πσ*- hyperconjugation. The constant χ (equals to 2.6805 × 10^{−3} mol^{1/3} at the pressure P = 0.1 MPa) does not depend on the nature of the considered liquids.

The choice of a monomolecular layer as a system whose properties determine the surface tension [_{b}, the surface tension becomes depending on the interaction of the upper surface layer with the nearest one [

Since in the present work we study the surface tension of boiling n-alkanes, one can expect that this quantity is also determined by the inter-molecular attraction within monomolecular surface layer. It is also clear that the electronic nature of this formation can differ from that of the above considered near-boiling liquids with molecular πσ*-hyperconjugation for which the values ofσ were found to be related to their molecular electronic structure.

Earlier it was shown that σ for different liquids (including the boiling ones) can be described by the following equation (see Equation (5) from [

σ = ε 0 ( 1 − T / T c ) n (2)

where n is the number of surface liquid molecules that must be removed from their equilibrium positions in order to increase the initial liquid surface by one-unit area. Further, ε_{0} is the absolute value of the average one-particle zero-point binding energy describing any separate molecule of the surface liquid layer at T = 0 K.

It was also shown [_{0} is proportional to the absolute value of the one-particle energy of intermolecular dispersion attraction |E_{d}|:

ε 0 = κ | E d | = φ N | E o r b | . (3)

In its turn, |E_{d}| is directly proportional [_{orb}|, where E_{orb} is the energy of those MOs which contribute essentially to dispersion attraction. Usually, the given MOs are the HOMOs or the nearest to the HOMOs occupied molecular orbitals [

Before determining the number n in Equation (2) one should consider some structural features of the surface monolayer of liquid n-alkanes. So, for the wide temperature range of liquid n-alkanes it has been shown experimentally [

Let us consider in more detail the formation of a new unit surface area of a liquid n-alkane. It can be modeled by removing a straight circular cylinder with a unit base area from its surface layer. The choice of just such a cylinder as the volume to be removed is due to the requirement of a minimum of the total energy necessary to overcome the dispersion attraction between the molecules from its side surface and the neighboring molecules outside the cylinder.

The energy of this intermolecular attraction is proportional to the number of molecules situated on the cylinder’s side surface, and therefore, to its side surface area. Therefore, a cylinder with the smallest side surface, namely a straight circular cylinder, will correspond to the minimal total attractive energy.

Since the base of the cylinder is a unit area circle, its radius r is equal to one π^{−1/2}. Then the value of its side surface area S equals to 2πrL or 2π^{1/2}L (where L is the thickness of the surface layer). If the area of one n-alkane molecule is s, that the number n in Equation (2) can be determined by the expression:

n = S / s = 2 π 1 / 2 L s − 1 . (4)

Then, using Equations (3) and (4) we can write Equation (2) for surface tension in the form:

σ = 2 π 1 / 2 φ N | E o r b | ( 1 − T b / T c ) L s − 1 (5)

Further, it is to be noted that for the molecules of n-alkanes their carbon atoms are spatially shielded by their hydrogen ones. This circumstance means that the dispersion attraction between the molecules within the liquid surface layer arises mainly due to the H∙∙∙H interactions of the neighboring molecules. Therefore, for the studied n-alkanes the number N in Equation (5) is equal to the total number of the hydrogen atoms entering the above considered occupied MO with its orbital energy E_{orb}. One can expect that the dispersion attraction will be the strongest if this MO has the following properties. Firstly, the number of the hydrogen atoms entering it should be as high as possible. Secondly, all the hydrogen atomic orbitals (AOs) belonging to this MO must have the same sign (+ or −). Thirdly, the given MO has to be either the highest occupied MO (HOMO) or one of the occupied MOs which is energetically close to the HOMO [

In the second part of the present work we investigate the QSAR between the n-alkanes electronic structure and the rate constants k in their reactions with hydroxyl radicals:

R 1 - ( CH 2 ) -R 2 + HO • = R 1 - ( C • H ) -R 2 + H 2 O (i)

Here R_{1} and R_{2} are linear alkyl substituents (or one hydrogen atom together with one methyl substituent in the case of ethane).

In order to find the sought expression connecting the rate constants k of the studied reactions (i) with the particularities of n-alkanes’ electronic structure we use the Eyring equation from the transition state theory [

k = [ τ ( k B T ) / h ] exp ( Δ S / R ) exp ( − Δ H / R T ) . (6)

In Equation (6) k_{B} and h are the Boltzmann and Planck’s constants, respectively, τ is the transmission coefficient of the reactions (i), ΔS and ΔH are, respectively, the entropy and the enthalpy of activation of the transition states for the reactions (i), T is the reaction absolute temperature, and R is the universal gas constant. Further, using the explicit form of ΔH, Equation (6) can be written as:

k = { [ τ ( k B T ) / h ] exp ( Δ S / R ) exp ( − P Δ V / R T ) } exp ( − Δ E / R T ) (7)

where ΔV is the change of the volume accompanying the formation of the transition state, ΔE is the activation energy, and P is the normal atmospheric pressure (P = 0.1 MPa = const).

Moreover, our DFT calculations have shown that the structures of the transition states of all the reactions (i) have some common spatial domain in which hydrogen atom transfer is being realized (see below). Therefore, it is reasonable to assume that the quantities τ, ΔS, and ΔV in Equation (7) are constant for all the studied n-alkanes. Since the absolute temperature of the reactions (i) is also constant (T = 297 ± 2 K), so we can present Equation (7) in the following form:

k = A exp ( − Δ E / R T ) , (8)

where A denotes the expression in its curly braces; therefore, A is constant too. In the logarithmic form Equation (8) may be written as:

log k = log A − Δ E / R T = C + D Δ E , (9)

where C = logA and D = −1/(RT) are some constant coefficients.

Further, as the absolute values of the atomic charges on the hydrogen atoms of the n-alkanes are near to zero, so the reactions (i) belong to the type of orbital-controlled reactions. To estimate the quantity ΔE we have used the expression given by Klopman in his work [

Δ E = [ 2 ( β c i c j ) 2 ] / ( E i − E j ) . (10)

In Equation (10) E_{i} is the energy of the molecular orbital of the hydroxyl radical, on which its unpaired electron is located, E_{j} is the energy of the SMO (E_{j} = E_{orb}, see above) for each investigated alkane, c_{i} and c_{j} are the LCAO coefficients of the atoms entering the H-atom transfer domain, and β is the resonance integral between the corresponding atomic orbitals. Therefore, one can assume that the value 2(β·c_{i}c_{j})^{2} in Equation(10) remains constant for all the n-alkanes under study. In this case, Equation (9) for reaction rate constants has the following final form:

log k = C + Z / ( E i − E o r b ) , (11)

where the constant Z = D ⋅ [ 2 ( β ⋅ c i c j ) 2 ] .

The electronic structures of all the considered n-alkanes in their optimized conformations were calculated by the DFT (B3LYP) method [

In the present work we have studied the series of the n-alkanes presented in _{orb}. It follows from the fact that the HOMOs are spatially shielded by the hydrogen atoms of the methylene groups, and, therefore, these orbitals cannot contribute to the value of the surface tension (

At the same time, for all the studied n-alkanes, we found the such occupied specific MOs denoted earlier as the SMOs which satisfies the above criteria. They include (with the same sign) the 1s-AOs of the hydrogen atoms of all methylene groups, and their energies E_{orb} are close to the energies of the corresponding HOMOs. The corresponding HOMOs and the SMOs for n-propane, n-pentane,

n-hexane, and n-nonane are shown in

Further, for any investigated n-alkane molecule we consider that the number of its methylene groups (denoted as N_{m}) is equal to the number of its carbon atoms. It means that its two terminal methyl groups are considered hereafter as the corresponding methylene ones connected with the two end hydrogen atoms belonging to its hydrocarbon chain. Then, the number of the hydrogen atoms N entering the SMO of any n-alkane will be equal to 2N_{m}. Besides, since the length of one hydrocarbon chain is proportional to N_{m}, then the layer thickness L in Equation (5) will be also proportional to N_{m} due to the normal orientation of the molecules to the two planar sides of the surface monolayer [

L = δ N m . (12)

Further, we assume that the molecules of the surface layer do not leave it for some time, which is large enough to neglect their movement in the normal direction to the two boundary planes of the monolayer. However, these molecules can possess some libration motion changing the angle between their hydrocarbon chains and this normal. The amplitude A of these librations will be also proportional to N_{m} (if we pass from one n-alkane to another one):

A = λ N m . (13)

Using the principle of corresponding states and bearing in mind that we deal with the boiling liquids (whose reduced boiling temperatures are very close) we suppose that each of the proportionality coefficients δ and λ in Equations (12), (13) is the same for all the studied n-alkanes. We can also assume that the average one-particle area s in Equation (5) is approximately equal to the area of some rectangle with the sides L and 2A:

s = 2 A L = ω ( N m ) 2 , (14)

where the proportionality coefficient ω = 2δλ is constant for any of the studied n-alkanes.

Substituting the defined quantities L, s, and N = 2N_{m} into Equation (5), we obtain the following expression for the surface tension:

σ = [ 4 π 1 / 2 φ N m | E o r b | ( 1 − T b / T c ) δ N m ] × [ ω ( N m ) 2 ] − 1 = ξ | E o r b | ( 1 − T b / T c ) = ξ G (15)

Here the coefficient ξ includes the multiplier 4π^{1/2} and the constants φ, δ, and τ, and the quantity G = | E o r b | ( 1 − T b / T c ) . Here it is to be noted that the number of methyl groups N_{m} is not presented in the final Equation (15). Now we have to take into account the influence onσ of the two end hydrogen atoms which belong to the hydrocarbon chain and do not enter the number 2N_{m} (see above). These hydrogen atoms are on the different sides of the surface monolayer, and they enter the corresponding SMOs for all the molecules forming the surface monolayer (

σ = ξ G + ρ . (16)

To define the constants ξ and ρ we have studied the set of the boiling short chain n-alkanes presented in _{b}, T_{c} and σ were taken from [_{orb}| which are required for the calculation of G = | E o r b | ( 1 − T b / T c ) . The constants ξ andρ can be obtained from the linear regression equation obtained by the least square method:

σ × 10 3 = 0.0448 G + 0.8333 . (17)

The values of σ calculated by means of Equation (17) are also given in

N | Alkane | T (˚K) | T_{c}(˚K) | |E_{orb}| (kJ/mol) | G (kJ/mol) | σ × 10^{3} (N/m) | ||
---|---|---|---|---|---|---|---|---|

Exp. | Calc. | LOOCV | ||||||

1 | Ethane | 184.55 | 305.32 | 893.58 | 353.4580 | 16.31 | 16.67 | 16.99 |

2 | Propane | 231.11 | 369.83 | 873.56 | 327.6656 | 15.50 | 15.51 | 15.53 |

3 | Butane | 272.65 | 425.12 | 863.34 | 309.6393 | 14.93 | 14.71 | 14.67 |

4 | Pentane | 309.22 | 469.70 | 845.16 | 288.7624 | 14.07 | 13.77 | 13.72 |

5 | Hexane | 341.88 | 507.60 | 841.27 | 274.6572 | 13.41 | 13.14 | 13.12 |

6 | Heptane | 371.58 | 540.20 | 837.02 | 261.2699 | 12.45 | 12.54 | 12.57 |

7 | Octane | 398.83 | 568.70 | 835.47 | 249.5532 | 12.09 | 12.01 | 11.99 |

8 | Nonane | 423.97 | 594.60 | 833.73 | 239.2533 | 11.53 | 11.55 | 11.56 |

9 | Decane | 447.30 | 617.70 | 833.00 | 229.7927 | 10.75 | 11.13 | 11.29 |

*The experimental values of T_{b}, T_{c} and σ were taken from [

of one of the studied compounds from the regression analysis and the following calculation of the predicted quantity for the removed compound. It can be seen that the values of the surface tension in the last two columns of

The plot of the correlation between the experimental values ofσ × 10^{3} [N∙m^{−1}] and the calculated parameter G [kJ∙mol^{−1}] is presented in

First of all, it is to be noted, that our DFT calculations have shown, that, for all the studied n-alkanes, it is these SMOs which determine the values of their surface tension (Section 4.1), are also decisive in their interaction with hydroxyl radicals. So, ^{●}. This interaction generates a transition state characterized by the formation of a three-center C-H-O bond (see

In Section 4.2 it was found the formulae log k = С + Z / ( E i − E o r b ) ,which connects the rate constants k of the above reactions (i) and the energies of the corresponding SMOs. In order to find parameters C, Z, and E_{i} which are common for all the n-alkanes, we have used the experimental values of k taken from the works [_{i} are equal to 8.1254, 496.25 kJ/mol, and 1003.13 kJ/mol, respectively. The values of logk calculated by means of Equation (11)

N | Alkane | k_{exp}_{.} × 10^{12 } (cm^{3}∙mol^{−1}∙s^{−1}) | −logk_{exp.} | |E_{orb}| (kJ·mol^{−1}) | −logk_{calc}_{.} |
---|---|---|---|---|---|

1 | Ethane | 0.222 | 12.65 | 893.58 | 12.66 |

2 | Propane | 1.12 | 11.95 | 873.56 | 11.95 |

3 | Butane | 2.10 | 11.68 | 863.34 | 11.68 |

4 | Pentane | 5.30 | 11.28 | 845.16 | 11.27 |

5 | Hexane | 5.91 | 11.22 | 841.27 | 11.19 |

6 | Heptane | 7.26 | 11.14 | 837.02 | 11.11 |

7 | Octane | 8.24 | 11.08 | 835.47 | 11.09 |

8 | Nonane | 9.36 | 11.03 | 833.73 | 11.05 |

9 | Decane | 9.90 | 11.00 | 833.00 | 11.04 |

with the above parameters are presented in the last column of _{calc} practically coincide with the corresponding experimental ones. Moreover, these values logk_{calc} are in a rather good agreement with the latest experimental data [

For the series of short chain n-alkanes we have identified the occupied Specific Molecular Orbitals (SMOs) which play a decisive role in determining their various physical and chemical properties: the surface tension σ near their normal boiling points and the reaction rate constants k with hydroxyl radicals (at 297 ± 2 K). Basing on the theoretical consideration of the intermolecular dispersion attraction in the surface layers and the orbital controlled interactions for the reactions R 1 - ( CH 2 ) -R 2 + HO • = R 1 - ( C • H ) -R 2 + H 2 O , the two quantitative relationship were received. They connect the energy values |E_{orb}| of these SMOs with the above quantities σ and k. The calculated values of σ and k are in fairly good agreement with the corresponding experimental data.

Obviously, the results of the present work can be also applied, for example, to cyclo-alkanes and other organic compounds, reacting with free radicals other than OH^{●}. It can also be assumed that the revealed role of the SMOs in the aforementioned dispersion attraction may serve for understanding of the nature of evaporation and condensation of the hydrocarbons under consideration. These problems will be considered in our next works.

The authors declare no conflicts of interest regarding the publication of this paper.

Gorbachev, M.Y., Gorinchoy, N.N. and Arsene, I. (2021) Key Role of Some Specific Occupied Molecular Orbitals of Short Chain n-Alkanes in Their Surface Tension and Reaction Rate Constants with Hydroxyl Radicals: DFT Study. International Journal of Organic Chemistry, 11, 1-13. https://doi.org/10.4236/ijoc.2021.111001