In order to consider the above property of the studied n-alkanes (from CH₄ to n-C₁₀H₁₂) we shall use the van’t Hoff isochore equation (see [4] [5]) written in its logarithmic form:

log 10 S = A + B Δ H R T = A + B ′ Δ H . (4)

Here S is solubility of a chemical compound (in our case any of the considered n-alkanes). The quantity S is usually expressed in molar fraction [4] and [5]. ΔH is the partial molar enthalpy connected with formation of a saturated aqueous solution for each of the studied n-alkanes. R is the gas constant. A and B' are some constant quantities depending on the nature of the considered n-alkane. T is the absolute temperature at that the dissolution of the n-alkanes takes place. As we consider the solubility of the n-alkanes at the constant T (T = 298.15 K) and at the normal atmospheric pressure P (P = 0.1 MPa), so we can include the quantities R and T into the constant B, forming some new constant B' (B' = B/RT).

Further, since for the studied n-alkanes their S values are of the order of 10⁻⁴ mol∙L⁻¹ [6], we consider them rather small in the sense that they lead to insignificant changes in volume during the formation of joint “alkane-water” systems. In other words, these volume changes are small enough to consider the dissolution process to be isochoric and described by the van’t Hoff isochore equation. Thus, for the quantity ΔH we can write the following approximate equality:

Δ H = Δ E + P Δ V ≈ Δ E (5)

Here ΔE is the partial molar energy change which accompanies the formation of the aqueous solution for any of the n-alkanes under study.

Now we have to define the quantity ΔE. In order to do it, we shall take into account the following reasons. At first, let us consider one n-alkane molecule which belongs to the studied aqueous solution and is surrounded by water molecules from all sides. We can consider that this molecule creates some cavity in the surrounding aqueous medium. Obviously, the creation of the given cavity requires some amount of energy ΔE and, thus, it is an endothermic process. Let, further, the quantity of the surface of this cavity be equal to Q. Then the quantity ΔE₁ can be defined as some work which is done against the forces responsible for the surface tension σ of water at the considered temperature T. In this case we can write for ΔE₁ the following equation:

Δ E 1 = σ Q (6)

Further, in the present work we shall consider that the value of Q is proportional to the number n_m of the methylene groups entering the molecular structure of the studied n-alkane. It should be noted here that the molecular structure of all the n-alkanes under study (including methane) are considered as consisting of a certain number of methylene groups and two terminal hydrogen atoms. So, we can rewrite Equation (6) as:

Δ E 1 = c σ n m (7)

Here c is some coefficient of proportionality between Q and n_m, which is supposed constant for all the n-alkanes.

Now we have to define the second addendum ΔE₂ in the total energy quantity ΔE. If the first addendum ΔE₁ describes (see above) the formation of the one- particle cavity in an aqueous medium containing the considered n-alkanes, then, in its turn, ΔE₂ is the energy addendum to ΔE, which is being defined by the intermolecular interaction between the MOs of the considered n-alkane molecule and the MOs of the water molecules surrounding it and forming its aqueous shell. It has been shown in the recent work [7] that all the O-H bonds of the above water molecules are situated within the surface of the given shell. In its turn, some part of the lone electronic pairs of these water molecules is directed towards the given n-alkane molecule. Thus, based on these data, in the present work we suppose that the above interaction is being realized by means of interaction of some specific unoccupied molecular orbitals of the n-alkanes and the MOs belonging to the water molecules of the mentioned aqueous shell and describing some number n_p of the lone electronic pairs of these water molecules.

Further, let the energy of the above-mentioned specific MO of the n-alkane molecule be E_orb and the number of the hydrogen atoms, whose 1s-AOs enter the given SMO with great values of their molecular-orbital coefficients, be n_H. Then, based on the data of the work [3], we can consider that the energy effect, describing the interaction of the n-alkane molecule with the surrounding water molecules will be proportional to the product |E_orb| ∙ n_H, or E_orb ∙ n_H (here we write E_orb instead of |E_orb|, based on the above-mentioned virtual character of the SMO of the n-alkane molecule; it means that its energy is positive and, therefore, |E_orb| = E_orb).

At the same time, the quantity ΔE₂ will be proportional to both the just considered product E_orb ∙ n_H and the above-mentioned number n_p. Thus, we can write for the quantity ΔE₂ the following equation:

Δ E 2 = χ E o r b n H n p , (8)

where the coefficient of proportionality χ is supposed to be constant for all the n-alkanes under study. In its turn, the number n_p is proportional to the surface Q of the aqueous shell, surrounding the considered n-alkane molecule, and Q is proportional to the number n_m of its methylene groups (see Equations (6) and (7) above), so we can rewrite Equation (8) in the final form:

Δ E 2 = γ E o r b n H n m , (9)

where γ is some constant coefficient of proportionality for all the studied n- alkanes.

Now, using the one-particle quantities ΔE₁ and ΔE₂ (see Equations (7) and (9), which fit only for one n-alkane molecule under consideration, we can write the following equation:

Δ E = N A ( Δ E 1 + Δ E 2 ) = c N A σ n m + γ N A E o r b n H n m , (10)

which holds for the molar quantity ΔE (see Equation (5)). In Equation (10) N_A is Avogadro number. Further, using Equations (5) and (10), Equation (4) can be presented as:

log 10 S = A + c N A B ′ σ n m + γ N A B ′ E o r b ⋅ n m n H , (11)

or in the following final form:

log 10 S = A + ξ n m + μ E o r b ⋅ n m n H . (12)

In Equation (12), the constants ξ and μ include the corresponding constants from Equation (11) and do not depend on the nature of the n-alkanes under study. The quantity of the surface tension of water σ is included in ξ, as it is also constant (as T is constant, T = 298.15 K).

In order to reveal the above relationship, it seems reasonable that the new equation which is similar to Equation (1) (received in the work [1]) fits also for the new reaction type of the studied shot chain n-alkanes with the new reagent: nitronium cation. Really, Equation (1) holds for the calculation of the reaction rate constants describing the elimination of hydrogen atoms from molecules of the studied n-alkanes by means of hydroxyl radicals. At the same time, in the case of nitronium cations, the hydrogen atom transfer from n-alkanes to NO 2 + occurs simultaneously with the corresponding electron transfer from the same SMO which earlier contained the electron of the given hydrogen atom (see the work [8]). Therefore, the transfer of the corresponding hydride anion H⁻, whose electrons have left the same SMO of the n-alkane, occurs through the NO 2 + ⋯ H ⋯ C three-center bond. This conclusion is also confirmed by our DFT calculations performed for the [n-C_nH_2n+1]⁺ systems. Thus, the same SMOs are responsible both for the abstraction of a hydrogen atom in the case of the reaction of n-alkanes with hydroxyl radicals [1] and for the transfer of the hydride anion in the reaction of n-alkanes with nitronium cations.

In order to verify Equation (12) received in the previous Section we have investigated the n-alkanes presented in Table 1. Beginning from ethane, all these compounds possess linear hydrocarbon chains. The geometry optimization of all the considered systems shows that their molecules have the planar carbon chains in which any four consecutive carbon atoms are in the trans-configuration. The subsequent analysis of the virtual MOs of the given n-alkanes (see Section 2.2) show that their LUMOs satisfy all the above-mentioned criteria for their consideration as the sought SMOs. Indeed, these LUMOs contain 1s-AOs of all the hydrogen atoms and, moreover, with one common sign (“+” or “−”) of the corresponding MO coefficients. For the case of methane, propane, n-pentane, and n-octane their LUMOs (their SMOs) are shown in Figure 1.

In order to demonstrate that namely the considered virtual orbitals are responsible for the interaction of n-alkanes with neighboring water molecules, we have calculated the interaction between the two molecules, namely, n-pentane and water. The results of this DFT calculation are presented in Figure 2. One can see that there is a clear-cut interaction between the lone electronic pairs of the oxygen atom of the water molecule and the SMO (the LUMO) of the n-pentane one.

For each of the studied n-alkanes, Table 1 contains their quantities n_m, n_H, and E_orb, along with the experimental values −log₁₀S taken from the work [6].

Further, the numerical values of the constants A, ξ, and μ (see Equation (12)) were defined by means of the least square method applied to the data set in Table 1. The usage of the found values of A, ξ, and μ in Equation (12) allows one to write it in the following final form:

− log 10 S = 0.4220 + 0.4198 n m + 4.28287 × 10 − 5 E o r b n m n H . (13)

The last but one column of Table 1 contains the values of −log₁₀S of the studied n-alkanes calculated by means of Equation (13). The last column of Table 1 includes the values of −log₁₀S which were obtained by the use of the so-called “Leave-one-out” Cross-Validation Scheme (LOOCV). The given statistical method consists of consequent removal of one of the investigated compounds from the regression analysis and subsequent calculation of the predicted quantity for the removed compound.

The comparison of the values of −log₁₀S presented in the last three columns of Table 1 shows that these values are very close to each other and to the corresponding experimental ones. It means that Equation (13) (and, therefore, Equation (12)) describes the solubility of the studied n-alkanes in water adequately.

In our previous work [1] we have investigated the rate constants for the reaction of shot chain n-alkanes (C₂ - C₁₀) with hydroxyl radicals in the gaseous phase at T = 298.15 ± 2 K and P = 0.1 MPa. Unlike [1], in the present paper, we consider the rate constants for the reaction of some n-alkanes with nitronium cations in the HNO₃-93 wt% H₂SO₄ solution at 293 K (see the work [13]). Thus, in the present work we deal with the new type of the reaction of n-alkanes: their electrophilic nitration which is being realized in the liquid phase (instead of the earlier considered gaseous one).

Since we consider the electrophilic nitration of the n-alkanes by means of nitronium cations, their occupied SMOs determined in [1] will interact with a certain virtual molecular orbital of nitronium cation. The given MO of NO 2 + can be characterized, firstly, by its negative energy (because of the electrophilic nature of nitronium cations) and, secondly, by its significant spatial extent. Our DFT calculations show that the MO of nitronium cation with the absolute value of its energy 702.88 kJ∙mol^−1, fits both the above-mentioned requirements. Figure 3 demonstrates the spacial shape of this MO along with that of the SMO of n-pentane, and the MO of the optimized joint system [n-C₅H₁₂ & NO 2 + ], arising due to the interaction between nitronium cation and n-pentane.

As noted in Section 2.2, the rate constants for the reactions of alkanes with NO 2 + can be calculated by Equation (1), in which E_i is the energy of the just discussed unoccupied MO of the studied cation. This redefinition of E_i will also lead to a change in the values of C and Z in Equation (1). In order to determine the new values of C and Z, which are constant for all the studied n-alkanes, we have used the experimental values of their reaction rate constants k with NO₂⁺ [13]. The new C and Z were defined by means of the least square method. Thus, for the rate constants of the studied nitration reaction, Equation (1) has the following final form:

− log 10 k = 5.5360 + 664.59 702.88 − | E o r b | . (14)

Table 2 contains the experimental values of k [13] and −log₁₀k, along with the earlier calculated (and taken from [1]) values of E_orb of the studied n-alkanes. It contains also the values of −log₁₀k calculated by means of Equation (14). The last column of Table 2 includes the values of −log₁₀k calculated by means of the LOOCV scheme (see above). The comparison of the values −log₁₀k presented in the three last columns of Table 2 shows that for each of the studied n-alkanes the corresponding values of −log₁₀k are very close to each other. Thus, Equation (14) describes the rate constants for the electrophilic nitration of the considered n-alkanes adequately.

It is to be noted that in this work we limited ourselves to consideration of n-alkanes, since, as our calculations show, when passing to branched systems, significant changes in the SMOs (the LUMOs) occur, which, in turn, depend on the branching site of the corresponding molecules.