Detailed kinetic study of the ring opening of cycloalkanes by CBS-QB3
  calculations
Detailed Kinetic Study of the Ring Opening  of Cycloalkanes by CBS-QB3 Calculations
Detailed Kinetic Study of the Ring Opening  of 
Cycloalkanes by CBS-QB3 Calculations 
B. Sirjean,1 P.A. Glaude,1 M.F. Ruiz-Lopez,2 R. Fournet1,*
1) Département de Chimie Physique des Réactions, UMR 7630 CNRS-INPL, ENSIC, 1, rue Grandville, 
BP 20451, 54001 Nancy Cedex, France 
2) Equipe de Chimie et Biochimie Théoriques, UMR CNRS-UHP No. 7565, Université Henri Poincaré 
– Nancy I, Boulevard des Aiguillettes, BP 239, 54506 Vandoeuvre-lès-Nancy, France 
e-mail:  Rene.Fournet@ensic.inpl-nancy.fr
Received date  
Title running head: Ring Opening of Cycloalkanes 
Corresponding author footnotes: Tel: 333 83 17 50 57, Fax : 333 83 37 81 20 
ABSTRACT 
This work reports a theoretical study of the gas phase unimolecular decomposition of cyclobutane, 
cyclopentane and cyclohexane by means of quantum chemical calculations. A biradical mechanism has 
been envisaged for each cycloalkane, and the main routes for the decomposition of the biradicals formed 
have been investigated at the CBS-QB3 level of theory. Thermochemical data (∆H°f, S°, C°p) for all the 
involved species have been obtained by means of isodesmic reactions. The contribution of hindered 
rotors has also been included. Activation barriers of each reaction have been analyzed to assess the 
mailto:Rene.Fournet@ensic.inpl-nancy.fr
energetically most favorable pathways for the decomposition of biradicals. Rate constants have been 
derived for all elementary reactions using transition state theory at 1 atm and temperatures ranging from 
600 to 2000 K. Global rate constant for the decomposition of the cyclic alkanes in molecular products 
have been calculated. Comparison between calculated and experimental results allowed to validate the 
theoretical approach. An important result is that the rotational barriers between the conformers, which 
are usually neglected, are of importance in decomposition rate of the largest biradicals.  Ring strain 
energies (RSE) in transition states for ring opening have been estimated and show that the main part of 
RSE contained in the cyclic reactants is removed upon the activation process.          
KEYWORDS: cyclobutane, cyclopentane, cyclohexane, biradicals, unimolecular initiation, CBS-QB3, 
theoretical calculations, rate parameters, thermochemical data, ring strain energy. 
1. Introduction  
Many important gas-phase or heterogeneous industrial processes such as combustion, partial 
oxidation, cracking or pyrolysis exhibit a complex chemical scheme involving hundreds of species and 
several thousands of reactions. In these thermal processes, cyclic hydrocarbons, particularly 
cycloalkanes, represent an important class of compounds. These molecules are produced during the gas-
phase processes though they can also be present in the reactants in large amounts; for example, a 
commercial jet fuel contains about 26% of naphtenes and condensed naphtenes, while in a commercial 
diesel fuel, this percentage reaches 40 % [1]. Modeling their reactivity currently represents an important 
challenge in the formulation of new fuels, less polluting and usable with new types of combustion in 
engines like “Homogeneous Charge Compression Ignition” (HCCI) [2]. During combustion or pyrolysis 
processes, cycloalkanes can lead to the formation of a) toxic compounds or soot precursors such as 
benzene (by successive dehydrogenations) and b) linear unsaturated species such as buta-1,3-diene or 
acroleïn (by the opening of the ring). Several experimental and modeling studies have been carried out 
on the oxidation of cyloalkanes in gas phase [3-9]. However, the modeling of the combustion of 
cycloalkanes remains difficult due to the lack of both, kinetic data for elementary reactions and 
thermodynamic data (∆H°f, S°, C°p) for the relevant species.  
A number of experimental and theoretical works have been reported on the decomposition of small 
cycloalkanes, such as cyclopropane [10-16]. Ring opening in this molecule is a well-known process and 
the rate of dissociation leading to propene formation has been extensively measured and calculated. Not 
less than 37 estimations of the rate constant are available on the NIST chemical kinetics database [17].  
Thermal decomposition of cyclobutane has been experimentally studied too and rate constants for the 
ring opening leading to the formation of two ethylene molecules have been measured [15-16, 18-19]. 
Theoretical studies on cyclobutane have mainly focused on the reverse reaction, namely, the 
cycloaddition of two ethylene molecules [20-23], since it represents a prototype reaction for the 
Woodward-Hoffmann rules and illustrates the usefulness of orbital symmetry considerations. Moreover, 
Pedersen et al. [24] have showed the validity of the biradical hypothesis by direct femtosecond studies 
of the transition-state structures. Theses studies have highlighted the fact that cycloaddition of two 
ethylene molecules can proceed through two different routes: one involves a tetramethylene biradical 
intermediate, while the other implies a concerted reaction that directly leads to cyclobutane formation. 
However, the latter reaction has been shown to have a high activation energy due to steric effects (see 
for instance ref [20] and therein) and to be much less favorable than the biradical process.  
Even though the study of ring opening of cyclopropane and cyclobutane are interesting from a 
theoretically point of view, larger cycloalkanes such as cyclopentane or cyclohexane are mainly 
involved in usual fuel. In spite of that, the unimolecular initiation of these molecules has been little 
investigated. Ring opening of cyclopentane and cylohexane has been experimentally studied by Tsang 
[25-26] who has reported the main routes of decomposition of these molecules. The mechanism and 
initial rates of decomposition were determined from single-pulse shock-tube experiments. For 
cyclohexane, a reaction pathway leading to the formation of 1-hexene has been considered only whereas 
for cyclopentane, the processes leading to either 1-pentene or to cyclopropane + ethylene have both 
been investigated, in accordance with the products experimentally detected. These results have been 
confirmed by further experimental studies by Kalra et al. [27] and Brown et al. [28]. In addition, Tsang 
[25] has shown that the experimentally obtained global rate parameters are consistent with a biradical 
mechanism for ring opening (Scheme 1): 
Scheme 1. Biradical mechanism for cyclopentane ring opening [25] 
Tsang connected the global rate parameters for the decomposition of cyclopentane in 1-pentene (k1) 
or in cyclopropane+ethylene (k2) to the rate parameters of the elementary reactions shown in Scheme 1. 
The equilibrium constant Keq=kb/kc has been estimated by the group additivity method proposed by 
Benson [29] and using some analogies with reactions of linear alkanes. From these assumptions, the rate 
constants of the elementary reactions have been estimated. The same procedure has been applied for the 
rate of decomposition of cyclohexane in 1-hexene. Even if the rate parameters obtained for the 
elementary reactions are consistent with the formation of a biradical, no transition state has been defined 
for validating the suggested mechanism. Moreover, the route leading to ethylene and trimethylene 
through C-C bond cleavage is rather ambiguous since other pathways are considered by Tsang, as for 
instance the direct decomposition of the n-pentyl biradical into cyclopropane and ethylene (Scheme 1). 
All of these studies showed that biradicals are central to the understanding of reaction mechanisms as 
well as to the predictability of reaction products and rates in the ring opening of cycloalkanes. 
Our aim in the present work is to analyze the ring opening of cyclobutane, cyclopentane and 
cyclohexane by means of high level quantum chemical calculations in order to obtain accurate rate 
constants for elementary reactions. Comparison of the computed rate constants with available 
experimental data allows us to validate the theoretical approach. We explore several plausible pathways 
that could be involved in the decomposition of the biradicals and we discuss the evolution of the ring 
strain energy (RSE) in going from the reactants to the transition state for ring opening. 
2. Computational method  
Quantum chemical computations have been performed on an IBM SP4 computer with the Gaussian03 
(G03) software package [30]. The high-level composite method CBS-QB3 [31] has been used. Analysis 
of vibrational frequencies confirmed that all transition structures (TS) have one imaginary frequency. 
Intrinsic Reaction Coordinate (IRC) calculations have been systematically performed at the B3LYP/6-
31G(d) [32-33] level to ensure that the computed TSs connect the desired reactants and products.  
Only singlet biradicals states have been considered and their study at the composite CBS-QB3 level 
requires two specific comments. First, at this level, geometry optimization of the systems is performed 
by density functional theory (DFT) using an unrestricted B3LYP method and a CBSB7 basis set. It is 
worth noting that the use of one determinantal wavefunction to describe open shell singlet biradicals can 
be questionable. However, previous studies have shown that the geometries obtained in this way 
compare well with those obtained at more refined computational levels [34-38]. Second, in CBS-QB3 
calculations, a correction for spin-contamination in open-shell species is added to the total energy. It 
includes a term of the form ( )2200954.0 thSSE −−=∆      where  denotes the expected value of the 2S 2Ŝ  
operator and 2thS  the corresponding theoretical value (e.g. 0 for a singlet state). This correction was 
derived for systems displaying a small spin-contamination. However, because of strong singlet-triplet 
mixing in the unrestricted wavefunction for biradicals, the 2S  values are close to 1 and this leads to a 
systematic error in the CBS-QB3 energy correction of about 6 kcal.mol-1. Several papers have pointed 
out this limitation [39-41] and some authors have proposed to remove the spin-contamination correction 
in this case. In the present work, we have preferred to use an empirical parameter specifically designed 
to handle spin-contamination in singlet biradicals so that singlet-triplet gaps for hydrocarbons biradicals 
are correctly described at the CBS-QB3 level. All energy values for singlet biradical species are 
therefore corrected by an expression of the form: 
∆Espin = -0.031 (S2-S2th)                     (1) 
with S2th =1. 
3. Thermochemical data  
Thermochemical data (∆Hf°, S°, Cp°) for all the species involved in this study have been derived from 
energy and frequency calculations and are collected in Table 1. In the CBS-QB3 method, harmonic 
frequencies, at the B3LYP/CBSB7 level of theory, are scaled by a factor 0.99. Explicit treatment of the 
internals rotors has been performed with the hinderedRotor option of Gaussian03 in accordance with the 
work of Ayala and Schlegel [42]. A systematic analysis of the results obtained was made in order to 
ensure that internals rotors were correctly treated. Thus, for biradicals a practical correction was 
introduced in order to take into account the symmetry number of 2 for each CH2(•) terminal group. This 
symmetry is not automatically recognized by Gaussian in the case of a radical group. Moreover, it must 
be stressed that in transition states, the constrained torsions of the cyclic structure have been treated as 
harmonic oscillators and the free alkyl groups as hindered rotations. 
Enthalpies of formation (∆Hf°) of species involved in this study have been calculated using isodesmic 
reactions [43] excepted for cyclanes and 1-alkene for which more accurate experimental enthalpies of 
formation [44] can be found in the literature. Thanks to the conservation of the total number and types 
of bonds, very good results can be obtained due to the cancellation of errors on the two sides of the 
reaction. Several isodesmic reactions have been considered for the calculation of ∆Hf°  in order to obtain 
an average value. However, results appear to be strongly dependent on the accuracy of the experimental 
data used for species involved in isodesmic reactions, especially for biradicals.   
Table 1. Ideal gas phase thermodynamics properties obtained by CBS-QB3 calculation.  is 
expressed in kcal.mol
KfH 298,∆
-1 and  and are given in cal.molo KS298 )(TCp
° -1.K-1. 
      )(TCp
°     
Species 
KfH 298,∆
KS298 300 K 400 K 500 K 600 K 800 K 1000 K 1500K P.G. 
 12.74 56.78 13.28   18.03   22.25   25.74   31.04   34.92   40.98 D3h
 6.79 63.19 16.97 23.47 29.37 34.33 41.94 47.48 55.94 D2d
 -18.27 70.21 20.99   29.17   36.67   43.01   52.81   59.92   70.74 D5h
 -29.43 71.55 25.45   35.27   44.28   51.96   63.89   72.55   85.66 D3d
 -22.99 74.09 25.61   35.33   44.30   51.95   63.84   72.49   85.60 D2
 0.82 78.27 24.52   32.14   38.82   44.39   53.03   59.41   69.39 C1
 -4.40 84.87 29.11   39.91   47.71   54.90   65.51   73.16   84.94 C1
 -5.09 79.85 25.99   32.74   38.83   44.05   52.37   56.64   68.53 C1
 -9.90 91.75 31.26   39.41   46.75   53.05   63.11   70.69   82.62 C1
 26.31 66.32 18.06   22.98   27.35   31.02   36.58   40.52   46.70 C2h
 25.41 78.56 23.57   29.58   34.92   39.42   46.46   51.70   59.91 C2
 67.43 79.01 22.95   27.80   32.16   35.90   41.92   46.54   53.94 C1
 68.13 77.33 23.59   28.36   32.50   36.10   41.93   46.46   53.82 C2h
62.84 87.11 27.87   34.29   40.04   44.93   52.73   58.64   68.05 C1
 62.46 87.12 28.37   34.50   40.11   44.95   52.71   58.62   68.03 C1
 62.79 89.54 28.15   34.26   39.89   44.75   52.52   58.44   67.90 C1
 58.50 98.74 31.50   39.48   46.76   52.98   62.83   70.22   81.87 C1
59.41 94.29  33.61   41.17   48.03   53.96   63.47   70.70 82.15 C1
59.08 92.96 34.02   41.97   49.00   54.97   64.47   71.58   82.74 C1
Hence, for these systems, ∆Hf° has been obtained from the prototype reaction: 
•R1-•R2 + 2R3H  → HR1-R2H  +  2 •R3,               (2) 
where •R1-•R2 represents the biradical and •R3 = •H, •CH3, •C2H5 or n-•C3H7. 
The computed entropy of cyclopentane has been corrected in order to take into account the 
experimental symmetry of the molecule (D5h). Indeed, CBS-QB3 calculations lead to a non planar 
geometry with  C1 symmetry. In addition, a low frequency of 22 cm-1 can be associated with a puckering 
motion of the ring. According to Benson [29], this pseudo-rotation is so fast that cyclopentane can be 
treated as dynamically flat (with a symmetry number σ=10). Thus, we corrected the entropy of 
cyclopentane by R ln10 (equation 3): 
Sc-C5H10 = S c-C5H10(CBS-QB3) – R ln 10               (3) 
The computed value, 70.0 cal.mol-1.K-1, is in good agreement with the experimental value [29].  
To our knowledge, no experimental enthalpy of formation for biradicals •C4H8•, •C4H10• or •C6H12• 
has been reported. However, we can compare our values with those obtained from an estimation based 
on bond dissociation energy (BDE) according to reaction (4): 
CnH2n+2 = •CnH2n• + 2 H•                      (4) 
where CnH2n+2 represents a linear free alkane and •CnH2n• is the corresponding biradical. ∆H°r for 
reaction 4 corresponds to twice the BDE for a C-H bond and the enthalpy of formation of •CnH2n• can 
therefore be calculated from equation (5): 
∆H°f (•CnH2n•, 298K) = 2BDE + ∆H°f (CnH2n+2, 298K) - 2 ∆H°f (H•, 298K)       (5) 
This calculation rests on the assumption that no interaction exists between the two radical centers. 
Table 2 compares estimated values using equation (5) and CBS-QB3 results for the most stable 
conformation of the biradicals •C4H8•, •C4H10• or •C6H12• . The BDE value for a primary C−H bond in 
a linear alkane has been taken equal to 100.9 kcal.mol-1, as proposed by Tsang [45] (it corresponds to 
C-H bond dissociation in propane to give the n-propyl radical). Experimental enthalpies of formation for 
the molecules used in equation (5) come from NIST webbook [44]. As shown in Table 2, a very good 
agreement is obtained between CBS-QB3 calculations using isodesmic reactions and values estimated 
by BDE and equation (5). This result corroborates the consistency of the electronic calculation scheme 
for the biradicals, in particular, the use of a broken symmetry method to optimize their geometry, as 
discussed in Section 2. 
Table 2. Comparison of ∆H°f (in kcal.mol-1, at 298 K) estimated from equation (5) and from theoretical 
calculations.  
Biradical ∆H°f from 
equation (5) 
This work 
•C4H8• 67.57 67.43 
•C5H10• 62.52 62.46 
•C6H12• 57.64 58.50 
4. Kinetic calculations 
The rate constants involved in the mechanisms were calculated using TST [47]: 
( ) ⎟
⎛ ∆−⎟
Trpdk buni   exp  exp 
   κ                (6) 
where and ≠∆S ≠∆H  are, respectively, the entropy and enthalpy of activation and rpd is the reaction 
path degeneracy. For reactions involving H-transfer, a transmission coefficient, namely κ(T), has been 
calculated  in order to take into account tunneling effect. We used an approximation to κ, provided by 
Skodje and Truhlar [47]: 
( ) ( )[ ]
( ) ( )[ ]
)/sin(
     
  for  
      
  for 
                (7) 
where Tkh B 
1   ,
)Im( 
2  ==
In equation (7), ν≠ is the imaginary frequency associated with the reaction coordinate, ∆V≠ is the zero-
point-including potential energy difference between the TS structure and the reactants, and V is 0 for an 
exoergic reaction and the zero-point-including potential energy difference between reactants and 
products for an endoergic reaction. 
The calculation of ≠∆H  was performed by taking into account electronic energies of reactants and 
TSs but also the enthalpy of reaction (Scheme 2). 
Reactant
Product
Scheme 2.Calculation of enthalpy of reaction: ≠∆H  
Thus, ≠∆H (reactant → product) is calculated from the following relation: 
∆H≠ R → P( ) =  ∆H1≠ (elec) +  ∆H−1≠ (elec) +  ∆Hr( ) 2            (8) 
and ≠∆H (product → reactant) by : 
∆H≠ P → R( ) =  ∆H1≠ (elec) +  ∆H−1≠ (elec) −  ∆Hr( ) 2              (9) 
where  and  are, respectively, the enthalpy of activation for the direct and reverse 
reactions, calculated from electronic energies, ZPVE and thermal correction energy. 
)(1 elecH
≠∆ )(1 elecH
rH∆ corresponds to 
the enthalpy of reaction estimated using isodesmic reactions [48]. The use of equations (8) and (9) 
ensures the consistency between kinetic and thermodynamic data and  improves the activation enthalpy 
results since the values obtained for rH∆  are more accurate than direct electronic calculations. 
The kinetic data are obtained with a fitting of equation 6 in the temperature range 600-2000K, with the 
following modified Arrhenius form:  
k = A Tb exp (-E/RT)                      (10) 
5. Ring opening mechanisms 
The ring opening mechanisms for cyclobutane, cyclopentane and cyclohexane are now discussed. In 
all schemes presented below, electronic free enthalpies are reported in kcal.mol-1 and are relative to the 
reference cycloalkane at P=1 atm. The value in bold corresponds to free enthalpy at T=298 K while the 
value in italics corresponds to T=1000 K.  
5.1 Cyclobutane 
Scheme 3 presents the global mechanism obtained for the ring opening of cyclobutane. In this scheme, 
we only considered conformers (molecules, biradicals, TSs) of lowest free enthalpies. Especially, we do 
not account for conformations of the biradical (2) (gauche and trans) and we only considered the 
gauche conformer since its free enthalpy is lowest. This point will be discussed in more detail below.    
100.5
0.0 -15.7
-18.0
-22.9
2 (s) 57.8
(s) singlet state
∆G°(T)  in kcal.mol-1  (bold : T= 298K, italic : T=1000K)  
Scheme 3. Mechanism obtained for the ring opening of cyclobutane and for conformers of lowest 
energies. 
The ring opening of cyclobutane involves a low free enthalpy of activation; ∆G≠ = 60.7 kcal.mol-1 at 
298 K which corresponds to an activation enthalpy (∆H≠) of 62.7 kcal.mol-1.  This value is lower than 
that involved in the dissociation of a C-C bond in the corresponding linear alkane (∆H≠ ≈ 86.3 
kcal.mol-1) [49]. This result will be discussed below but we can anticipate that it is mainly due to the 
high ring strain energy involved in the cyclobutane molecule, that is partially removed in the transition 
state. Following ring opening, three ways of decomposition for tetramethylene have been investigated. 
The route leading to the formation of two ethylene molecules corresponds to the most favorable one, as 
already mentioned in the literature [15-23]. The process leading to 1-butene represents another possible 
pathway though this elementary reaction involves a high stressed cyclic transition state for H-transfer 
(four-member ring) and displays a high activation energy ( ≠∆H =17.6 kcal.mol-1 vs 2.8 kcal.mol-1 for 
the β-scission leading to C2H4, at 298 K). This pathway, however, might provide a non–negligible 
contribution to the total process. Indeed, C4H8 has been identified as a minor product in experimental 
studies of cyclobutane decomposition [18]. The third route consists of the abstraction of two hydrogen 
atoms to form buta-1,3-diene and H2. The reaction step has a high activation barrier ( ≠∆H = 40 
kcal.mol-1 at T=298K) and should not compete with the previous mechanisms.  
A concerted reaction leading from cyclobutane to direct formation of two ethylene molecules has 
been envisaged but no consistent TS could be obtained. As mentioned previously, several studies [20-
23] have been performed on the reverse reaction, i.e. the cyclic addition of two ethylene molecules. The 
authors found that a two-step biradical-type reaction is expected to be favored (by about 24 kcal.mol-1) 
over concerted pathways. Sakai [20] estimated activation energies of 77.6 kcal.mol-1 and 58.8 kcal.mol-1 
for the concerted and stepwise processes, respectively, at the MP2//CAS/6-311+G(d,p) level. However, 
it is important to underline that this author reported a second-order saddle point structure (two negative 
eigenvalues) for the concerted mechanism, which therefore does not correspond to a transition state. 
Table 3 gives the kinetics parameters of the modified Arrhenius form (equation 10) for all the 
elementary processes involved in Scheme 3. Unimolecular initiation of cycloalkanes is unimportant for 
low temperatures in thermal processes and we only give kinetics parameters for T > 600K in all the 
tables. 
Table 3. Rate parameters for the unimolecular initiation of cyclobutane at P=1 atm, 600 ≤ T (K) ≤ 
2000 K and related to scheme 3. 
 k1-2 k2-1 k2-C2H4 k2-4 k2-5
log A (s-1) 18.53 12.21 7.32 5.57 2.23 
n -0.797 -0.305 1.443 2.171 2.995
(kcal.mol-1) 
64.85 1.98 3.03 16.44 37.61
  To the best of our knowledge, no previous quantum calculations have been performed to estimate rate 
parameters for the elementary reactions involved in the thermal decomposition of cyclobutane. In 1971, 
Beadle et al. [19] experimentally studied the pyrolysis of cyclobutane and reported rate parameters for 
the ring opening of c-C4H8 (k1-2 and k2-1) and for the decomposition of the biradical in C2H4 (k2-C2H4). 
Their estimations are based on tabulated thermochemistry and additivity methods. The A factor and 
activation energy proposed by Beadle et al. are as follows: 3.63 1015 s-1 and 63.34 kcal.mol-1 for k1-2, 
2 1012 s-1 and 6.6 kcal.mol-1 for k2-1, and 1.17 1013 s-1 and 8.25 kcal.mol-1 for k2-C2H4. In spite of 
considerable uncertainty in that study, as mentioned by the authors, their results are in good agreement 
with ours. Thus, at 800 K, the ratio between our value and that proposed by Beadle et al. is 1.7, 2.0 and 
0.7 for k1-2, k2-1, and k2-C2H4, respectively. Moreover, it is worth noting that our CBS-QB3 quantum 
calculations provide an accurate energy for the cycloaddition of two ethylene molecules (formation of 
biradical 2). The free enthalpy of activation obtained is equal to 56.3 kcal.mol-1 at 298K, which 
corresponds to an activation energy ∆H≠ = 47.7 kcal.mol-1. This last value can be compared with the 
activation energy of 56.8 kcal.mol-1 reported by Sakai [20] at the MP2/CAS/6-311+G(d,p) level though 
this author obtained  a second order saddle point.  
 As mentioned previously, gauche/trans interconversion of (2) has been neglected in Scheme 3. 
However, the β-scission reaction involves a low ∆G≠ (5.1 kcal.mol-1 at 298K) that could be of the same 
order of magnitude than the rotational barrier around the central CC bond. Accordingly, it can be 
interesting to consider the two conformations of the biradical for this particular but important pathway. 
A detailed scheme is presented in Scheme 4. 
1 2 (s) 3 (s)
6.66.6
46.30.0 48.4
-22.9
-22.9
56.2 53.5
56.0 55.7
Gauche Trans
(s) singlet state
∆G°(T)  in kcal.mol-1  (bold : T= 298K, italic : T=1000K)  
Scheme 4. Detailed mechanism of C2H4 formation from the ring opening of cyclobutane and obtained 
by considering different conformers of C4 biradicals. 
 Cyclobutane ring-opening leads to the gauche biradical (2). The latter can decompose in two C2H4 
molecules or rotate to give the trans conformation (3), which in turn, can also decompose in two C2H4 
molecules. The kinetic parameters for all processes involved in Scheme 4 are collected in Table 4. 
Table 4. Rate parameters for the unimolecular initiation of cyclobutane at P=1 atm, 600 ≤ T (K) ≤ 2000 
K and related to scheme 4. 
 k1-2 k2-1 k2-3 k3-2 k2-C2H4 k3-C2H4
log A (s-1) 18.53 12.21 11.30 11.27 7.63 7.32 
n -0.797 -0.305 0.461 0.545 1.453 1.521 
Ea (kcal/mol) 64.85 1.98 4.29 3.26 4.79 2.07 
Tables 3 and 4 give the kinetic parameters involved in the thermal decomposition of cyclobutane. In 
order to validate our results, we compare the global rate constant for the process:  
Cyclobutane → C2H4 + C2H4,    
measured experimentally by Barnard et al. [15] and Lewis et al. [16], with that derived from our 
computations (Schemes 3 and 4). We consider the quasi-stationary state approximation (QSSA) for 
biradicals. For Scheme 3, QQSA applied to the gauche biradical leads to the simple expression:  
HCscheme
=                          (11) 
For scheme 4, QSSA leads to a more complex expression:  
  k  
scheme
g             (12) 
with , 
           
4242424242 322323323122312
HCHCHCHCHC kkkkkkkkk k
−−−−−−−−−−
The fit of the global rate constants obtained from relations (11) and (12) leads to the rate parameters 
presented in table 5. 
Table 5. Rate parameters for the global reaction cC4H8 → 2 C2H4, at P=1 atm, 600 ≤ T (K) ≤ 2000 K 
and related to Schemes 3 and 4. 
 3scheme
4scheme
log A (s-1) 20.29 21.52 
n -1.259 -1.606 
Ea (kcal/mol) 67.69 68.16 
Figure 1 compares our results (  and ) with the absolute value measured directly by 
Barnard et al. [15] and by Lewis et al. [16] for the thermal rate decomposition of cyclobutane in two 
ethylene molecules.  
3scheme
4scheme
Figure 1. Comparison between calculated rate constant and experimental data for the global reaction 
cC4H8 → C2H4 + C2H4
As shown in Figure 1, our results are close to the experimental values, validating the consistency of the 
theoretical approach. Computed rate constants are always slightly lower than those obtained by Lewis et 
al. [16] (maximum factor 2.4) or by Barnard et al. [15] (maximum factor 4.3). It is interesting to note 
that differences between  and  decrease with temperature, which is consistent with 
rotational hindrance (20% at 600K and 8% at 2000 K). Though these differences are weak, the rate 
4scheme
3scheme
constant obtained by explicit consideration of the two conformations of the biradical (trans and gauche) 
is closer to experimental results than the rate constant calculated from Scheme 3.  
5.2 Cyclopentane 
Scheme 5 shows the global mechanism obtained for the ring opening of cyclopentane. As for 
cyclobutane, the global mechanism does not take into account the different conformations of the 
biradical •C5H10• and we only consider the conformation with the lowest free enthalpy.  
Due to weaker ring strain energy, the free enthalpy of activation of the ring opening of cyclopentane 
(∆G≠ = 80.5 kcal.mol-1 at 298 K) is higher than that obtained in cyclobutane (∆G≠ = 60.7 kcal.mol-1 at 
298 K).  
4 (s)
100.5
117.3
102.7
107.1
(s) singlet state
∆G°(T)  in kcal.mol-1  (bold : T= 298K, italic : T=1000K)  
Scheme 5. Mechanism obtained for the ring opening of cyclopentane and for conformers of lowest 
energies. 
Four routes of decomposition have been investigated for the biradical 4, the most favorable one being 
the formation of 1-pentene by H-transfer. This last reaction exhibits an activation energy much lower 
than the equivalent process in cyclobutane ( ≠∆H = 6 kcal.mol-1 vs 17.6 kcal.mol-1 for cC4H8 at 298 K).  
This noticeable difference can be ascribed to different strain energy in the corresponding transition 
states. Conversely, the β-scission process leading to ethylene and cyclopropane is much more difficult 
than that leading to ethylene in cyclobutane. In this last case, the presence of two radical centers in 
(•C4H8•) in β position weakens the inner C-C σ-bond. Another interesting point concerns the 
dehydrogenation reaction of biradical 4 leading to penta-1,4-diene (6) and H2. At low temperature 
(298K), this reaction is competitive with β-scission (reaction 4 →7) but becomes unimportant at high 
temperature (1000K) due to a low change of entropy between the biradical and TS. Finally, the reaction 
of biradical (4) to yield ethyl-cyclopropane (8) is unlikely to occur at either room or high temperature 
because it involves a high stressed cyclic transition state for H-abstraction and always displays quite 
high activation energy. For completeness, it must be noted that formation of a trimethylene biradical 
from 4 has been envisaged. However, the singlet state for this system could not be obtained, geometry 
optimizations leading systematically to the formation of cyclopropane (only the triplet state could be 
optimized). This may be due to the monodeterminantal character of the methodology used for the 
geometry optimizations but it has been shown that ring closure of trimethylene to form cyclopropane 
occurs very fast anyway [50]. Table 6 gives the kinetics parameters of the modified Arrhenius forms 
(equation 7) for all the elementary processes involved in Scheme 5. As mentioned previously, just a few 
studies have been devoted to the kinetics of the thermal decomposition of cyclopentane [25, 27]. Tsang 
[25] measured by comparative-rate-pulse shock tube experiments, the ring opening of cyclopentane, and 
suggested rate expressions for k1-4, k4-1, k4-5 and k4-7, over the temperature range 1000 K – 1200 K. The 
estimations were based on experimental results obtained for the thermal decomposition of cyclopentane 
and on the thermochemical considerations methods proposed by Benson [29]. 
Table 6. Rate parameters for the unimolecular initiation of cyclobutane at P=1 atm, 600 ≤ T (K) ≤ 2000 
K and related to Scheme 5. 
 k1-4 k4-1 k4-5 k4-6 k4-7 k4-8
log A (s-1) 18.11  9.89 6.77 0.51 9.78 -3.07 
n -0.466 0.311 1.480 3.015 1.1 4.157 
Ea (kcal/mol) 85.18 1.7 7.76 17.78 26.16 32.43 
However, the analysis did not take into account transition state and Tsang used a constant value of 
0.16 for the ratio k4-5/ k4-1 in accordance with the disproportionation to combination ratio for n-propyl 
radicals. At 1100 K, the ratios obtained between our values and those proposed by Tsang for k1-4, k4-1, 
k4-5 and k4-7 are, respectively, 0.5, 1.5, 1.6 and 1.7. Despite uncertainties in both experimental and 
theoretical calculations, the agreement is therefore quite satisfying. It is also possible to compare some 
computed rate parameters for elementary reactions in Scheme 5 with values obtained using semi-
empirical relations as proposed by O’Neal [51]. Accordingly, the activation energy involved in reaction 
4 → 6, corresponding to a H-transfer (disproportionation), can be estimated by the following 
relationship [51]: 
Ea = ED  +  ESE                       (13) 
where ED represents the activation energy involved in the disproportionation of two alkyl free radicals 
and ESE represents the strain energy of the cyclic transition state. The latter was taken to be 6.3 
kcal.mol-1 since the transition state exhibits a five-member ring [29]. An activation energy of 1 
kcal.mol-1 can be reasonably taken into account for disproportionation [49].  From our calculations the 
activation enthalpy of reaction 4→ 6, is estimated to 6 kcal.mol-1 at 298 K and therefore it is in good 
agreement with the semi-empirical estimation. On the other hand , our quantum calculations give an 
activation energy equal to 26.6 kcal.mol-1 at 298 K for the β-scission of biradical 4 (reaction 4→7). This 
value can be compared with the mean activation energy for the β-scission of a C-C bond in an alkyl free 
radical (28.7 kcal.mol-1 [49]).  
In Scheme 5, biradicals 4 represents the most stable conformer of the biradical although it does not 
lead directly to the formation of 1-pentene.  Since some activation energies, such as that of the reaction  
4 → 5 ( ≠∆H = 6.7 kcal.mol-1 at 298 K), have values close to that of the rotation barrier heights, the role 
of rotational hindrance has been examined in the case of cyclopentane. Scheme 6 contains the detailed 
mechanism. 
2 (s) 3(s)
77.5 77.1
65.3 64.7
(s) singlet state
∆G°(T)  in kcal.mol-1  (bold : T= 298K, italic : T=1000K)  
Scheme 6. Detailed mechanism of 1-pentene formation from the ring opening of cyclopentane and 
obtained by considering different conformers of C5 biradicals. 
Two conformations of the biradical •C5H10• are involved in Scheme 6 but only one (biradical 3) leads 
to the formation of 1-pentene by H-abstraction. This result has been validated by IRC calculations. 
Geometries of TS3-5 and biradicals (2) and (3) are presented in Figure 2.  
As shown, the distance between carbon atom 2 and hydrogen atom 11 is larger in biradical 2 than in 
biradical 3 (compares structures a and c), which renders H-abstraction for this last species more 
favorable. This result can be explained by a gauche interaction in biradical 2 involving carbon atoms 4 
and 1. In biradical 3, carbon atoms 1, 3, 4 and 5 are in the same plan and no gauche interaction between 
carbon atoms 1 and 4 is possible.  
Figure 2. Geometries relative to biradical 2 (a), and TS  (b), biradical 3 (c) 
Figure 2. G
 It is wor
Indeed, in 
activation e
In cyclope
only one c
for the mec
Table 7. R
2000 K and
log A (s-1) 
(kcal/mol) 
In order 
rotational b
experiment
2.819 Å
eometries relative to biradical 2 (a), and TS3-5 (b), biradical 3 (c) 
th noting the different role played by rotational hindrance in cyclo
cyclobutane (Scheme 4), the rotational barrier is of the same o
nergy for β-scission and both conformers may lead to the formati
ntane, the rotational barrier is lower than the activation energy for
onformation (biradical 3) allows the latter process. Table 7 summ
hanism shown in Scheme 6. 
ate parameters for the unimolecular initiation of cyclopentane at
 related to Scheme 6. 
k1-2 k2-1 k2-3 k3-2 k3-5
18.11 10.74 10.97 10.54 6.89 
-0.466 0.207 0.569 0.602 1.494
84.76 1.7 2.68 2.97 7.45 
to validate the rate parameters involved in Schemes 5 and 6 and to
arriers between biradicals 2 and 3, we have compared the glob
ally by Tsang [25] for the following reactions: 
cyclopentane → 1-pentene,  k1= 1016.1 exp(-84840 (cal.mol-1
cyclopentane  → cyclopropane + ethylene,  k2= 1016.25 exp(-95060
2.645 Å
2.819 Å
2.645 Å
pentane and cyclobutane. 
rder of magnitude as the 
on of ethylene molecules. 
 1-pentene formation and 
arizes the rate parameters 
 P=1 atm, 600 ≤ T (K) ≤ 
 quantify the effect of the 
al rate constants obtained 
)/RT), 
 (cal.mol-1)/RT), 
with global rate constant estimated by QSSA performed on biradical 4 (Scheme 5) and biradicals 2 and 
3 (Scheme 6). The rate expressions are: 
54415 
51,   
kschemeg ,                       (14) 
533253122-312
5332216 
51,        
−−−−−
kkkkkk
kschemeg ,               (15) 
74415 
71,   
kschemeg ,                    (16) 
and the kinetics parameters are given in Table 8. 
Table 8. Rate parameters for the global reactions cC5H10 → 1-pentene and cC5H10→ cC3H6 +C2H4 at 
P=1 atm, 600 ≤ T (K) ≤ 2000 K and related to Schemes 5 and 6. 
scheme
gk −  
scheme
gk −  
scheme
gk −  
log A (s-1) 20.39 24.33 20.06 
n -0.970 -1.542 -0.878 
(kcal/mol) 
92.86 112.49 92.23 
Comparison of global rate constants estimated by the QSSA approach and those proposed by Tsang 
are presented in Figure 3. In the range 1000K –1200K, the values calculated from our global rate 
constants for the formation of 1-pentene (  and ) are in a good agreement with those 
obtained by Tsang, our values being lower by a factor 1.2 to 2.  
scheme
scheme
Figure 3. Comparison between calculated rate constant and experimental data for the global reactions 
c-C5H10 → 1-pentene and cC5H10 → c-C3H6 + C2H4
A similar result has been obtained in the case of the reaction leading to cyclopropane and ethylene, 
with values underestimated by a factor 0.7 to 2.5 with respect to Tsang estimations. Concerning the 
influence of rotational hindrance in the formation of 1-pentene,  is always found to be lower 
than  over the temperature range (28 % at 600K and 8% at 2000K). By analogy with 
cyclobutane, calculations performed by considering the different conformations for biradical •C
scheme
scheme
5H10• 
permit to obtain results closer to experiment. Moreover, the effect of rotational hindrance is slightly 
greater in the case of cyclopentane than cyclobutane. As said above, this can be explained by the fact 
that biradical •C5H10• must necessarily rotate in order to yield 1-pentene whereas in cyclobutane, both 
gauche and trans conformations of the biradical can decompose in two ethylene molecules. 
5.3 Cyclohexane 
Only a few numbers of studies have been performed on the unimolecular initiation mechanism of 
cylohexane [26, 28]. Scheme 7 shows our results for the ring opening of c-C6H12 and subsequent 
reactions. As before, in this scheme, we only consider the lowest free enthalpy conformers of the 
biradicals. Table 9 summarizes the computed rate parameters.  
5 (s)
106.5 127.0
110.8
8 94.7106.9
53.287.5
C4 pathway
(s) singlet state
∆G°(T)  in kcal.mol-1  (bold : T= 298K, italic : T=1000K)  
Scheme 7. Mechanism obtained for the ring opening of cyclohexane and for conformers of lowest 
energies. 
Table 9. Rate parameters for the unimolecular initiation of cyclohexane at P=1 atm, 600 ≤ T (K) ≤ 2000 
K and related to scheme 7.  
 k1-5 k5-1 k2-5 k5-2 k5-6 k5-7 k5-8 k5-c-C3H6
log(A s-1) 21.32 9.91 20.11 10.38 2.46 -1.33 10.40 5.23 
n -0.972 0.136 -0.785 0.137 2.569 3.800 0.994 2.185 
Ea (kcal/mol) 92.63 2.09 85.77 2.13 1.42 17.22 25.75 44.25 
In our study, the chair and boat conformations of cyclohexane have been taken into account. As 
mentioned by Dixon and al. [52], the boat structure (C2v symmetry) is a transition state that connects the 
chair structure with a twist boat (D2 symmetry) conformation. We did not locate the transition state 
corresponding to the C2v boat structure but we have obtained a D2 twist boat conformation that does 
correspond to a characterized energy minimum. Hereafter, this twist boat conformation will be simply 
refereed to as the “boat conformation” (symmetry number of 4, D2 symmetry). At high temperature, the 
concentration of this boat conformation cannot be neglected and the equilibrium constant Keq, 
corresponding to the reaction: 
c-C6H12 (chair)          c-C6H12 (boat) ,      
has been fitted in the 600 - 2000 K  temperature range using the relation: 
K rreq 271.1
3265exp  )    ( exp  +−=+−=
           (17) 
CBS-QB3 calculations performed from chair and boat conformations of cyclohexane and considerations 
of isodesmic reactions have permitted to calculate and . The mean values for and  
are, respectively, 6.5 kcal.mol
°∆ rS
°∆ rH
°∆ rH
°∆ rS
-1 and 2.5 cal.mol.l-1. From equation (17), Keq is equal to 0.047 at 753K 
that corresponds to 95.5 % of cyclohexane molecules in chair conformation. This result is in very good 
agreement with the value reported by Walker and Gulati [53] who reported a slightly larger value 
(99.5%). Another estimation of the equilibrium constant has been reported by Eliel and Wilen [54] from 
experimental values obtained at 1073K. They found that 25% of the twist boat structure is present in the 
mixture, that leads to Keq = 0.33. From equation (17), we obtain Keq = 0.17 at 1073K  that is consistent 
with the estimation made by Eliel and Wilen. Thus, it appears that, in the range of temperature 
considered in our study, the chair/boat conformation ratio must be taken into account in kinetic 
calculations.  
Two different TSs have been found for the ring opening of cyclohexane depending on its initial boat 
or chair structure (Figure 4), the TS corresponding to the former conformation being 2.4 kcal.mol-1 
lower in free energy. Since activation energy for ring opening is much higher than the energy involved 
in boat/chair conformation change, we conclude that only the lowest TS should be taken into account in 
the kinetic scheme.   
a)   b)   
Figure 4: TS obtained from the ring opening of cyclohexane with the boat conformation (a) and the 
chair conformation (b) 
The free enthalpy of activation obtained for the ring opening of cyclohexane (∆G≠ = 85.5 kcal.mol-1, 
at T = 298K) is higher than that obtained for cyclobutane (∆G≠ = 60.7 kcal.mol-1) or cyclopentane (∆G≠ 
= 80.5 kcal.mol-1) at the same temperature, and close to the value found for a linear alkane. This result 
will be discussed in detail below but it is interesting to point out that it is consistent with an “unstrained 
structure” in cyclohexane. The main way of decomposition for the biradical •C6H12• (biradical 5) is the 
formation of 1-hexene, as previously mentioned by Tsang [26]. Indeed, this process involves a slightly 
constrained transition state for H-abstraction (six-member ring) with a low activation energy ≠∆H = 1.8 
kcal.mol-1 (vs 17.6 kcal.mol-1 for c-C4H8 and 6 kcal.mol-1 for c-C5H10 at 298 K). The value of ≠∆H = 1.8 
kcal.mol-1 can be compared with the semi-empirical estimation of Ea given by equation 13 and 
corresponding to a disproportionation process. Taking 1 kcal.mol-1 for ED and 0.7 kcal.mol-1 for ERS 
(disproportionation of two alkyl radicals for ED and a ring strain energy of a six-member ring for ERS), 
Ea = 1.7 kcal.mol-1, in very good agreement with our calculation.  Owing to this very low activation 
energy, other routes for •C6H12• decomposition are unlikely. An interesting remark can be made for 
β-scission leading to the formation of •C4H8• and ethylene (reaction 5 → 8). In the experimental study 
performed by Tsang [26], no cyclobutane was detected what could implicitly be explained by a very low 
reaction rate for •C4H8• formation compared to 1-hexene. Analyzing free enthalpies of activation at 
T=1000K in Scheme 7 shows that the ratio between the disproportionation leading to 1-hexene (reaction 
5→ 6) and the β-scission (reaction 5→ 8) is about 120. A comparable value is obtained in cylopentane 
decomposition (Scheme 5). However, in the case of the thermal decomposition of cyclopentane, 
cylopropane is detected experimentally [25] (though it represents a minor product) whereas for 
cylohexane, no cyclobutane is observed at all. This discrepancy can be explained by the very fast 
decomposition of the biradical •C4H8• in two ethylene molecules, compared to the cyclization reaction, 
as shown in Scheme 3. This assumption cannot be verified using the experimental data reported by 
Tsang [26] since ethylene formed by initiation would represent a minor part of the total concentration of 
this molecule, which is principally obtained in propagation reactions (decomposition of 1-hexene and 
cylohexyl radical).  Activation energy of the β-scission of biradical 5 (reaction 5→ 8) is equal to 25.2 
kcal.mol-1 at 298 K, which is consistent with the β-scission of a C-C bond in an alkyl free radical, with a 
corresponding activation energy of 28.7 kcal.mol-1 [50].  
  Let us now consider the effect of conformers of •C6H12•, that was ignored in Scheme 7. The 
activation energy involved in the formation of 1-hexene from the biradical •C6H12• is quite low ( ≠∆H = 
1 kcal.mol-1 at 298 K) and might be competitive with rotational barriers. In accordance with this 
assumption, we developed a detailed mechanism for the formation of 1-hexene involving the different 
conformations of the biradical (Scheme 8). 
3 (s)
4 (s)
87.288.43.9
67.1 67.0
6 13.3-1.7
(s) singlet state
∆G°(T)  in kcal.mol-1  (bold : T= 298K, italic : T=1000K)  
Scheme 8. Detailed mechanism of 1-hexene formation from the ring opening of cyclohexane and 
obtained by considering different conformers of C6 biradicals. 
Table 10. Rate parameters for the unimolecular initiation of cyclohexane at P=1 atm, 600 ≤ T (K) ≤ 
2000 K and related to scheme 8. 
 k1-3 k3-1 k2-3 k3-2 k3-6 k3-4 k4-3 k4-6
log A (s-1) 20.68  11.08 20.16 11.23 2.17 10.11 11.40 4.17 
n -0.799 0.117 -0.810 0.073 2.923 0.720 0.359 2.295 
Ea (kcal/mol) 92.44 0.70 85.99 0.81 0.77 2.50 2.98 0.20 
Table 10 summarizes the rate parameters for elementary reactions in Scheme 8. In order to compare 
the global rate constant obtained by Tsang for the thermal decomposition of cyclohexane in 1-hexene 
[26] to that obtained from Schemes 7 and 8, we performed quasi-stationary-state approximation (QSSA) 
on biradical 5 (Scheme 7) and biradicals 3 and 4 (Scheme 8). Scheme 8 leads to the following 
expressions for the reaction c-C6H12 → 1-C6H12 : 
kchair
scheme8 =  
k3−6 k1−3
k4−6 k3−4  k1−3
C k4−3 +  k4−6( )
             (18) 
kboat
scheme8 =  
k3−6 k2−3
k4−6 k3−4  k2−3
C k4−3 +  k4−6( )
              (19) 
with C =  
k3−4 k4−6
k4−3 +  k4−6
 +  k3−1 +  k3−2 +  k3−6               (20) 
At a given temperature, the global rate constant can be calculated from rate constants given by 
equations (18) and (19) and equilibrium constant in order to have the ratio of boat and chair 
conformations. Thus, we obtained the following expression for the global rate constant:  
scheme
scheme
boateq
scheme
chairscheme
)61( +
=−              (21) 
where Keq is the equilibrium constant obtained from equation (17). 
For the more general Scheme 7, QSSA applied on the biradical 5 leads to the following relations: 
kchair
scheme7 =  
 k1−5 k5−6
k5−1 +  k5−6
       and       kboat
scheme7 =  
k2−5 k5−6
k5−2 +  k5−6
               (22) 
By analogy with equation (21), the global rate constant for Scheme 7 can be the calculated by using the 
equilibrium constant:  
kg(1−6)
scheme7 =  
kchair
scheme7
1 +  Keq
Keq  kboat
scheme7
1 +  Keq
           (23) 
Rate parameters obtained by fitting and  for temperatures ranging from 600K to 2000K 
are presented in Table 11.   
kg(1−6)
scheme7 kg(1−6)
scheme8
Table 11. Rate parameters for the global reactions cC6H12 → 1-hexene at P=1 atm, 600 ≤ T (K) ≤ 2000 
K and related to Schemes 7 and 8. 
scheme
gk −  
scheme
gk −  
log A (s-1) 20.45 20.29 
n -0.685 -0.639 
Ea (kcal/mol) 93.01 94.52 
Figure 5 shows the comparison of the values calculated from and  (Table 11) and those 
obtained from the rate constant proposed by Tsang [26], for temperatures between 950K and 1100K, i.e. 
in the range of validity of Tsang’s study.  
kg(1−6)
scheme7 kg(1−6)
scheme8
Figure 5: comparison between calculated rate constant and experimental data [26] for the global 
reactions c-C6H12 → 1-hexene. 
The agreement between k  and the rate constant proposed by Tsang is rather satisfactory since 
the ratio 
g(1−6)
scheme8
kg(1−6)
scheme8
kTsang
lies between 2 and 2.6. Another point concerns the difference obtained between the rate 
constants and  shown in Figure 6. By neglecting rotational hindrance (Scheme 7), the 
global rate constant is overestimated by a factor 2 compared to , in the temperature range  
950 – 1100 K.  This difference can be explained by the low activation energy involved in the formation 
of 1-hexene compared to rotational barrier, but also by entropic effects due to the difference between 
entropy of  the “linear” biradical (biradical 5) and biradicals 3 and 4 (Table 1). This results shows that 
by considering only the lowest energy biradical conformation, the global rate constant is largely over-
estimated. 
kg(1−6)
scheme7 kg(1−6)
scheme8
kg(1−6)
scheme7 kg(1−6)
scheme8
5.4 Ring strain energies 
The results obtained for the ring opening of cycloalkanes into biradicals, show an increase of the 
enthalpy of activation going from cyclobutane to cyclohexane. These differences are mainly due to the 
change of ring strain energy when one goes from cycloalkanes to TS. To discuss the results obtained, 
we first calculated the ring strain energy of cycloalkanes by using the usual definition of RSE [55] 
(equation 24) : 
RSE = ∆HRS = Hcyclo – n HCH2                 (24) 
where Hcyclo and HCH2, represent, respectively, the electronic enthalpies of cycloalkane and of CH2 
fragment in a strain-free alkane. n represents the number of CH2 fragments in the cyclic species. HCH2 
has been calculated by difference between the enthalpy of n-hexane and n-pentane. Hcyclo and HCH2 have 
been obtained by considering the electronic energy, zero-point energy and thermal corrections to 
enthalpy given at a CBS-QB3 level of theory at 298K.  
RSE can also be obtained by using enthalpies calculated by isodesmic reactions. Thus, equation 24 can 
be rewritten as : 
RSE = ∆HRS = ∆fH°cyclo – n ∆fH°CH2               (25) 
where ∆fH°cyclo and ∆fH°CH2 correspond, respectively, to the enthalpy of cycloalkane formation obtained 
from isodesmic reactions and enthalpy of a CH2 group formation, obtained by the difference between 
the enthalpy of formation of n-hexane and n-pentane.  
The values obtained for cyclobutane, cyclopentane, cyclohexane (twist boat) and cyclohexane (chair) 
are summarized in Table 12.  
RSE obtained at a CBS-QB3 level from direct electronic energies and those obtained from isodesmic 
reactions are in a very good agreement with values proposed by Cohen [56] and based on group 
additivity method. 
Table 12. Ring strain energies (in kcal.mol-1) of cycloalkanes as calculated at the CBS-QB3 level of 
theory and  those proposed by Cohen [56] at  T=298K.  
Cycloalkane Cyclobutane Cyclopentane Cyclohexane 
(Chair) 
Cyclohexane 
(Twist Boat) 
RSE given by 
Electronic energies 
27.0 7.4 1.1 7.5 
RSE calculated from 
Isodesmic reactions 
26.8 7.5 1.0 7.5 
RSE from Cohen [56] 26.8 7.1 0.7 / 
It is now interesting to estimate the remaining part of RSE contained in the transition states during 
ring opening. This can be done by comparing the enthalpies of activation obtained for the ring opening 
of cycloalkanes with those obtained for the dissociation of free-strain linear alkanes and by assuming 
that the difference obtained between the cyclic species and the corresponding linear one is only due to 
the ring strain energy. If no difference is observed, one can conclude that no remaining RSE is 
contained in the TS. 
For a linear alkane, the bond dissociation energy (BDE) can be assimilated with the activation 
enthalpy since the recombination of the two free radicals is barrierless. BDE between two secondary 
carbon atoms have been estimated by CBS-QB3 methods for n-butane, n-pentane and n-hexane by 
means of reaction (26): 
CnH2n+2  →  x  •C3H7  +  y  •C2H5                  (26) 
where CnH2n+2 represents a linear alkane; x and y depend on the value of n.  
BDE corresponds to the enthalpy of reaction (24) and can be expressed by the following equation: 
BDE = ∆rH° = x HC3H7 + y HC2H5 – HCnH2n+2 ≈ ∆H≠             (27)   
where HC3H7, HC2H5, represent, respectively, electronic enthalpy of n-propyl and ethyl radicals, and 
HCnH2n+2, the enthalpy of n-butane, n-pentane or n-hexane.  
Calculations of BDE in equation (27) can be done using electronic enthalpies or enthalpies estimated 
from isodesmic reaction.Table 13 gives the results obtained in the two cases and the experimental 
values proposed by Luo [57]. 
Table 13: BDE (in kcal.mol-1) of two secondary carbon atoms for unstrained linear alkanes, at 298 K  
(C-C) 
CBSQ-B3 
Electronic energies 
CBSQ-B3 
Isodesmic reactions
Experimentals 
BDE  [57] 
n-Butane 88.8 87.0 86.8 
n-Pentane 89.6 87.9 87.3 
n-Hexane 90.5 88.9 87.5 
BDE calculated from direct electronic calculations are higher that experimental ones [57]. On the other 
hand, the values deduced from isodesmic calculations are closer to these recommended values and show 
the best accuracy obtained with isodesmic reactions. If we consider that BDE can be assimilated to 
enthalpy of reaction in the case of linear unstrained alkane, we can compare the activation energies 
obtained for the opening of cycloalkanes and those obtained by removing the RSE from the BDE of the 
corresponding linear free-strain alkane. Table 14 shows this comparison from isodesmic calculations. 
Table 14. Comparison of ∆H≠ (in kcal.mol-1), obtained for the ring opening of cycloalkanes and those 
estimated from linear unstrained alkane, at T=298K by taking into account isodesmic reactions.  
Species ∆H
≠ for 
cycloalkane 
Ring strain energy 
(RSE) 
∆H≠ calculated from the 
corresponding linear alkane 
Remaining 
Cyclobutane 61.7 26.8 87 – 26.8 = 60.2 1.5 
Cyclopentane 82.4 7.5 87.9– 7.5 =  80.4 2.0 
Cyclohexane 
(chair) 
89.5 1.0 88.9 – 1.0 = 87.9 1.6 
Cyclohexane 
(twist boat) 
83.0 7.5 88.9 – 7.5 = 81.4 1.6 
The remaining RSE shown in table 14 permits to conclude that almost all of the RSE is removed in the 
transition state of cyloalkanes, excepted, perhaps, for cyclohexane (chair) where isodesmic calculations 
show a slightly increase of the ring strain energy in the transition state. Figure 6 shows the three 
transition state involved in the ring opening of the cycloalkanes studied. 
c) b) a) 
3.533 Å 3.329 Å
2.789 Å 
Figure 6: TS involved in the ring opening of cyclobutane (a), cyclopentane (b) and cyclohexane (c) 
Dudev et al. [55] have showed, from ab initio calculations, that the ring strain energy of cycloalkanes 
can be explained by contributions of ring bond angles, bond lengths or dihedral angles, this last 
parameter reflecting nonbonded interactions. Thus, the ring strain energy remaining in the TS of 
cyclohexane can be explained by the fact that the six-member ring is forced to adopt an energetically 
unfavorable gauche conformation, whereas n-hexane can exist in a strain-free trans-trans-trans 
conformation. Moreover, they showed that the effects of ring bond angles on RSE decrease when the 
size of the cycle increases while an opposite effect is obtained for nonbonded interactions. In our study, 
the formation of TS from cyclobutane involves an increase of the ring bond angles which permits to 
remove a large part of RSE. Indeed, in the TS of Figure 6a, ∠C1C2C3 = ∠C2C3C4  =  108.4°  vs 88.6° in 
cyclobutane. For TS involved in the ring opening of cyclopentane (Figure 6b), the large increase of the 
bond length between C1 and C2 (3.329 Å in TS vs 1.545 Å in cyclopentane) associated with an increase 
of the ring bond angle ( °=∠=∠ 8.114    542531 CCCCCC in the TS vs 104.8° in the molecule), may explain 
a large part of the removal of RSE in the TS. In cyclohexane, the low value of RSE is maintained in the 
transition state, that can be explained by gauche interactions remaining in the TS.  
6. Conclusions 
While much work have been carried out on the thermal reactions of aliphatic hydrocarbons, the high 
temperature reactions of cyclic hydrocarbons has not been explored extensively. In this study, the ring 
opening of the most representative cyclic alkanes, i.e. cyclobutane, cyclopentane, and cyclohexane have 
been extensively explored by means of quantum chemistry. All the possible elementary reactions have 
been investigated from the biradicals yielded by the initiation steps. The thermochemical properties of 
all the species have been calculated with the high-level CBS-QB3 method. The inharmonic contribution 
of hindered rotors have been taken into account and isodesmic reactions have been systematically used 
for the evaluation of the enthalpies in order to minimize the systematic errors. The enthalpies of 
formation of the biradicals have been compared to data obtained with a semi-empirical method and 
show a very good agreement. 
For all the elementary reactions, the transition state theory allowed to calculate the rate constant. 
Three parameters Arrhenius expressions have been derived in the temperature range 600 to 2000 K at 
atmospheric pressure. Tunneling effect has been taken into account in the case of internal H transfers. 
Thanks to the Quasi Steady State Approximation applied to the biradicals, rate constants have been 
calculated for the global reaction leading directly from the cyclic alkane to the molecular products. 
These values have been compared with the few data available in the literature and showed a rather good 
agreement. The main reaction routes are the decomposition to two ethylene molecules in the case of 
cyclobutane and the internal disproportionation of the biradicals yielding 1-pentene and 1-hexene in the 
case of cyclopentene and cyclohexane, respectively. An important fact highlighted in this work is the 
role of the internal rotation hindrance in the biradical fate. Whereas the energy barriers between 
conformers are usually of low energy in comparison to the reaction barriers, all the energies are close in 
this case and taking the rotations between the conformers into account changes the global rate constant 
especially for the largest biradicals. The analysis of the variation of the ring strain energy has also 
showed that the larger part is removed when going from the cycloalkanes to the transition states. These 
last structure are close to be unconstrained with between 1 or 2 kcal remaining. 
ACKNOWLEDGMENT  
The Centre Informatique National de l’Enseignement Supérieur (CINES) is gratefully acknowledged 
for allocation of computational resources 
SUPPORTING INFORMATION AVAILABLE 
  The full list of author in ref 30, the structural parameters for all the species investigated in this study 
the frequencies, energies and zero point energies. This material is available free of charge via the 
Internet at http://pubs.acs.org.  
http://pubs.acs.org/
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