Radicals are the most important intermediate compounds in the processes of chemical reactions; however, the study of these short-lived compounds is extremely difficult due to their high reactivity. When describing the sequence of chemical transformations (during the synthesis or decomposition of compounds), the paths of their occurrence primarily depend on the energy properties (enthalpies of formation) of the radicals. The enthalpies of formation of radicals cannot be measured by experimental calorimetric methods (except for stable radicals), and they are usually estimated based on various indirect kinetic methods or using thermodynamic calculations. However, the determination of the enthalpies of formation of radicals is additionally complicated by the fact that, according to the model of N.N. Semyonov [1], the decay of molecule R1R2 according to the bond R1–R2 consists of splitting fragments R1 and R2 and converting them into radicals \({\text{R}}_{1}^{ \bullet }\) and \({\text{R}}_{2}^{ \bullet }{\text{:}}\) R1R2 → R1–R2 + ε = \({\text{R}}_{1}^{ \bullet }\) + \({\text{R}}_{2}^{ \bullet },\) where R1 and R2 are fragments of molecules, ε is the possible value of the change in the energies of fragments of molecules upon their transition to radicals (energy of rearrangement of radicals), and \({\text{R}}_{1}^{ \bullet }\) and \({\text{R}}_{2}^{ \bullet }\) are the radicals into which the studied compound decomposes. For a long time after the publication of [1], it was assumed that the values of ε for radicals are zero or do not exceed the errors in the enthalpies of formation of radicals. Modern data on the enthalpies of formation of substances and radicals, as well as energies of bond dissociation D(R1–R2), and bond strengths E(R1–R2) (average thermochemical bond energies) [2, 3], make it possible to determine the values of ε using the relation D(R1–R2) = E(R1–R2) + \({{{{\varepsilon }}}_{{{{{\text{R}}}_{1}}{{\;}}}}} + {{\;}}{{{{\varepsilon }}}_{{{{{\text{R}}}_{2}}{{\;}}}}}\) and chemical physics formulas relating the enthalpies of atomization, as well as the enthalpies of formation of atoms, substances, and radicals with bond energies. When calculating the values of ε, the values of the rearrangement energies ε, which depend only on the form of the radical \({\text{R}}_{{}}^{ \bullet }\), are used, and for monatomic fragments of molecules, the rearrangement energy is zero. For diatomic molecules, the bond dissociation energies coincide with the average thermochemical bond energies D(R1–R2) ≡ E(R1–R2). To calculate the values of ε in the enthalpy of atomization ΔatH, instead of the value E(R1–R2), its expression in terms of the bond dissociation energy is introduced:

$$\begin{gathered} D\left( {{{{\text{R}}}_{1}}--{{{\text{R}}}_{2}}{{\;}}} \right)-{{{{\varepsilon }}}_{{{{{\text{R}}}_{1}}{{\;}}}}}{{\;}}-{{{{\varepsilon }}}_{{{{{\text{R}}}_{2}}{{\;}}}}}, \\ {{{{\Delta }}}_{{at}}}H = {{\Sigma }}{{{{\Delta }}}_{f}}{{H}_{{at}}} - {{{{\Delta }}}_{f}}H^\circ \\ = {{\Sigma }}{{E}_{i}} = {{\Sigma }}{{E}_{{{{{\text{R}}}_{1}}}}} + {{\Sigma }}{{E}_{{{{{\text{R}}}_{2}}}}} + E\left( {{{{\text{R}}}_{1}}{{\;}}-{{{\text{R}}}_{2}}{{\;}}} \right) \\ = {{\Sigma }}{{E}_{{{{{\text{R}}}_{1}}}}} + {{\Sigma }}{{E}_{{{{{\text{R}}}_{2}}}}} + D\left( {{{{\text{R}}}_{1}}{{\;}}-{{{\text{R}}}_{2}}{{\;}}} \right) - {{{{\varepsilon }}}_{{{\text{R}}_{1}^{ \bullet }}}} - {{{{\varepsilon }}}_{{{\text{R}}_{2}^{ \bullet }}}}. \\ \end{gathered} $$
(1)

The sums of the bond energies in fragments of molecules, \({{\Sigma }}{{E}_{{{{{\text{R}}}_{1}}}}}\) and \({{\Sigma }}{{E}_{{{{{\text{R}}}_{2}}}}}\), necessary for the calculations are constant and do not depend on which molecule they correspond to. This can be seen, for example, after transforming the equation for the enthalpy of atomization of the CaHbOcNd compound. Then we get

$$\begin{gathered} {{{{\Delta }}}_{{at}}}H = a\left( {{{{{\Delta }}}_{f}}H_{{\text{C}}}^{^\circ }} \right) + b\left( {{{{{\Delta }}}_{f}}H_{{\text{H}}}^{^\circ }} \right) + c\left( {{{{{\Delta }}}_{f}}H_{{\text{O}}}^{^\circ }} \right) \\ + \,\,d\left( {{{{{\Delta }}}_{f}}H_{{\text{N}}}^{^\circ }} \right)\,\,{\text{ + }}\,{{{{\Delta }}}_{f}}H_{{{\text{R}}_{1}^{ \bullet }}}^{^\circ } + {{{{\Delta }}}_{f}}H_{{{\text{R}}_{2}^{ \bullet }}}^{^\circ } + {{{{\Delta }}}_{f}}H_{{{{{\text{R}}}_{1}}{{{\text{R}}}_{2}}}}^{^\circ } \\ = {{\Sigma }}{{E}_{{{{{\text{R}}}_{1}}}}} + {{\Sigma }}{{E}_{{{{{\text{R}}}_{2}}}}} + {{{{\Delta }}}_{f}}H_{{{\text{R}}_{1}^{ \bullet }}}^{^\circ } + {{{{\Delta }}}_{f}}H_{{{\text{R}}_{2}^{ \bullet }}}^{^\circ } \\ - \,\,{{{{\Delta }}}_{f}}H_{{{{{\text{R}}}_{1}}{{{\text{R}}}_{2}}}}^{0} - {{{{\varepsilon }}}_{{{\text{R}}_{1}^{ \bullet }}}} - {{{{\varepsilon }}}_{{{\text{R}}_{2}^{ \bullet }}}}. \\ \end{gathered} $$
(2)

After transformation (2), we get the equation

$$\begin{gathered} a\left( {{{{{\Delta }}}_{f}}H_{{\text{C}}}^{^\circ }} \right) + b\left( {{{{{\Delta }}}_{f}}H_{{\text{H}}}^{^\circ }} \right) + c\left( {{{{{\Delta }}}_{f}}H_{{\text{O}}}^{^\circ }} \right) + d\left( {{{{{\Delta }}}_{f}}H_{{\text{N}}}^{^\circ }} \right) \\ = {{\Sigma }}{{E}_{{{{{\text{R}}}_{1}}}}} + {{\Sigma }}{{E}_{{{{{\text{R}}}_{2}}}}} + {{{{\Delta }}}_{f}}H_{{{\text{R}}_{1}^{ \bullet }}}^{^\circ } + {{{{\Delta }}}_{f}}H_{{{\text{R}}_{2}^{ \bullet }}}^{^\circ } - {{{{\varepsilon }}}_{{{\text{R}}_{1}^{ \bullet }}}} - {{{{\varepsilon }}}_{{{\text{R}}_{2}^{ \bullet }}}}. \\ \end{gathered} $$
(3)

From (3) it follows that the sum \({{\Sigma }}{{E}_{{{{{\text{R}}}_{1}}}}} + {{\Sigma }}{{E}_{{{{{\text{R}}}_{2}}}}}\) consists only of constants related to R1 and R2:

$$\begin{gathered} {{\Sigma }}{{E}_{{{{{\text{R}}}_{1}}}}} + {{\Sigma }}{{E}_{{{{{\text{R}}}_{2}}}}} = a\left( {{{{{\Delta }}}_{f}}H_{{\text{C}}}^{^\circ }} \right) + b\left( {{{{{\Delta }}}_{f}}H_{{\text{H}}}^{^\circ }} \right) + c\left( {{{{{\Delta }}}_{f}}H_{{\text{O}}}^{^\circ }} \right) \\ + \,\,d\left( {{{{{\Delta }}}_{f}}H_{{\text{N}}}^{^\circ }} \right) - \left( {{{{{\Delta }}}_{f}}H_{{{\text{R}}_{1}^{ \bullet }}}^{^\circ } + {{{{\Delta }}}_{f}}H_{{{\text{R}}_{2}^{ \bullet }}}^{^\circ }} \right) + \left( {{{{{\varepsilon }}}_{{{\text{R}}_{1}^{ \bullet }}}} + {{{{\varepsilon }}}_{{{\text{R}}_{2}^{ \bullet }}}}} \right). \\ \end{gathered} $$
(4)

For example, the calculated sum of bond energies for a phenyl fragment is \(\Sigma {{E}_{{{{{\text{C}}}_{{\text{6}}}}{{{\text{H}}}_{{\text{5}}}}}}}\) = (5052.83 ± 0.03) kJ mol–1, for which thermochemical data from substances with different chemical properties are used: C6H6, C6H5CH3, C6H5OH, C6H5NO2, C6H5NH2, C6H5COOH, C6H5C6H5, C6H5I, C6H5Br, C6H5Cl, C6H5F. The enthalpies of formation of these compounds are taken from [4]; of atoms and radicals, from [5]; and of the phenyl radical \({{{\text{C}}}_{{\text{6}}}}{\text{H}}_{5}^{ \bullet }\), from [6]: (337.2 ± 1.3) kJ mol–1.

As an example, let us estimate the energy properties of cyclopropane and the energy of rearrangement of the cyclopropyl radical. The enthalpy of formation of cyclopropane in the gas phase is (53.3 ± 0.5) kJ mol–1 [4]; and of the cyclopropyl radical, (279.9 ± 10.5) kJ mol–1 [5]. The enthalpy of atomization of cyclopropane is 3404.9 kJ mol−1. Using the data of Tables 1 and 2 [7], we determine the voltage energy of cyclopropane En relative to the sum of hydrocarbon bonds from these tables (relative to the enthalpies of atomization of the corresponding alkanes), which is equal to En = 117.1 kJ mol–1. From Tables 1 and 2, according to the data on binding energy E(C2–Н) in saturated hydrocarbons, it is possible to estimate the bond energies \(E\left( {{\text{C}}_{2}^{/}~-{\text{C}}_{2}^{/}} \right)\) in cyclopropane: 314.2 kJ mol–1. Based on the enthalpies of formation of the cyclopropyl radical, hydrogen, and propane, we obtain the value of the bond dissociation energy D(C3H5–H), equal to 444.8 kJ mol–1, and the rearrangement energy of the cyclopropyl radical is 34.5 kJ mol–1. The data on the enthalpies of formation of cyclic hydrocarbons were obtained similarly (Table 3).

Table 1.   Values of average thermochemical energies of Ci–H bonds in saturated alkanes
Table 2. Values of average thermochemical energies of Ci–Cj bonds in saturated alkanes
Table 3. Values of voltage energy En, bond energy \(E\left( {{\mathbf{C}}_{2}^{/}~-{\mathbf{C}}_{2}^{/}} \right),\) rearrangement energy of radicals ε of cyclic hydrocarbons

One of the most efficient ways to calculate values E and ε is their joint calculation for bonds with the same nearest environment, for which it is realistic to assume that the bond energies are close to E. We used this method to calculate the rearrangement energy of the azide radical [8].

As an example of determining the rearrangement energies of the radicals of framework compounds based on a joint calculation, we can cite the calculations of the values of ε for the adamantyl-1 and adamantyl-2 radicals. The calculation for adamantyl-2 is quite simple, when for the estimate of ε, we can compare the value E(C2–Н) from Tables 1 and 2 (410.3 kJ mol–1) with the bond dissociation energy D(C2–Н) in adamantane, calculated according to Eq. (1) from the enthalpies of formation of adamantyl-2 (61.9 kJ mol–1 [5]), the hydrogen atom ((217.998 ± 0.006) kJ mol–1 [5]), and adamantane ((–132.3 ± 2.2) kJ ⋅ mol–1 [9]) in the gas phase. With such an assessment, the rearrangement energy of the adamantyl-2 radical is 1.9 kJ mol–1.

For the calculation of the prestructuring energy of adamantyl-2, we can choose the cyclohexyl radical and the cyclohexane molecule, and based on the data on their enthalpies of formation (for cyclohexane and cyclohexyl, it is (–123.4 ± 0.8) [4] and (75.3 ± 6.3) kJ mol–1 [5], respectively) and using the data of Tables 1 and 2 (E(C2–C2) = 353.4 kJ mol–1), from the value of the enthalpy of atomization, we find E(C2–H) = 409.9 kJ mol–1 in cyclohexane with the dissociation energy D(C2–H) of 416.7 kJ mol–1 and we get ε = 6.8 kJ mol–1 for cyclohexyl. The dissociation energy D(C2–H) in adamantane is 412.1 kJ mol–1. Taking the same values E(C2–Н) in cyclohexane (409.9 kJ mol–1) and adamantane, we obtain ε = 2.2 kJ mol–1for adamantyl-2.

Two options can also be used to calculate the ε value of adamantyl-1. First, we compare the bond dissociation energy D(C3–H) in adamantane with the value E(C3–Н) for hydrocarbons from Tables 1 and 2 (403.1 kJ mol–1). Value D(C3–H) in adamantane, obtained from the enthalpies of formation of adamantyl-1 (51.5 kJ mol–1 [5]), atomic hydrogen, and adamantane, was 401.8 kJ mol–1. The energy for the rearrangement of radical adamantyl-1 was obtained equal to –1.3 kJ mol–1.

In another version of the calculation, adamantane was compared with structures containing C3–H bonds. Unfortunately, the authors do not know the enthalpy of formation of cyclic unstrained structures with C3–H bonds and the corresponding radicals for the atom \({\text{C}}_{3}^{ \bullet },\) therefore, for comparison with adamantane and radical adamantyl-1, it is best to use trans-octahydronaphthalene and its radical in position \({\text{C}}_{3}^{ \bullet }.\) However, there are no data on the enthalpy of formation of such a radical in the literature. For alkyl radicals \({\text{C}}_{3}^{ \bullet }\), we can compare the enthalpies of formation of 2-methylpropane ((–134.2 ± 0.6) kJ mol–1 [4]) and the radical (CH3)3C ((48 ± 3) kJ mol–1 [5]), the difference between which is 182.2 kJ mol–1. For 2-methylbutane ((–153.6 ± 0.9) kJ mol–1 [4]) and (C2H5(CH3)2C (29 kJ mol–1 [5]), the difference in the values of the enthalpies of formation was 182.6 kJ mol–1. Comparison of the enthalpies of formation in the gas phase of adamantane ((–132.3 ± 2.2) kJ mol–1 [9]) and adamantyl-1 (51.5 kJ mol–1 [5]) gives a difference in the enthalpies of formation of 183.8 kJ mol–1, which is close to the two previous values (they coincide not only within the limits for the error of radicals, but even within the limits for the error of compounds from [4]). If we accept trans-octahydronaphthalene (decalin) for a bicyclic structure, the difference in the enthalpies of formation of this compound is (–182.1 ± 2.3) kJ mol–1 [4] and if we accept this difference as the same for the corresponding radical as for adamantane and its radical, then we obtain the value of the enthalpy of formation of trans-octahydronaphthalene-1-yl, equal to 1.7 kJ mol–1. This value makes it possible to determine the bond dissociation energy D(C3–H) in trans-octahydronaphthalene equal to 401.7 kJ mol–1. The bond energy E(C3–H) in trans-octahydronaphthalene derived from its enthalpy of atomization and binding energies (according to Tables 1, 2) is 408.2 kJ mol–1. The radical rearrangement energy found from these values of trans-octahydronaphthalene-1-yl was –6.5 kJ mol–1. If we accept E(C3–Н) in adamantane is the same in magnitude as in trans-octahydronaphthalene (408.2 kJ mol–1), then from D(C3–H) = 401.8 kJ mol–1 in adamantane, we determine ε = –6.4 kJ mol–1 for adamantyl-1. This value, obtained for the structure closest in composition, is preferable to the previous one, although they coincide within the error of the enthalpies of formation of the radicals.

In a similar way using data on the enthalpies of formation of the compounds [4] and their radicals [5] (Tables 1, 2), the energy of restructuring ε a number of framework radicals was calculated (Table 4). From the values of the rearrangement energies of the radicals of the framework structures, it can be concluded that the formation of radicals in carbon atoms C2 and C3 somewhat reduces the energy of radicals relative to hydrocarbon radicals, and the appearance of a radical particle in cuban and the corresponding rearrangement of interatomic bonds significantly strain the cuban framework and increase the energy of its radical.

Table 4. Values of rearrangement energy ε of polycyclic (framework) radicals