Entropy data are reported using different calculation methods for internal rotors on
Over the past decades advancements made in quantum chemistry, including the evolution of density functional theory (DFT) and compound methods, coupled with increasing computer processing power, have allowed for substantial advancements in estimating thermochemical and other chemical properties and systems [
Thermochemical properties including enthalpies, entropies, and heat capacities for stable molecules and for radicals corresponding to the loss of H atom through abstraction reactions are important and necessary for building and analyzing detailed chemical kinetic models. Several research studies have focused on linear and branched alkanes, including
Enthalpy, entropy, and heat capacity are all important because of the influence they have on reactions’ paths. The TΔS entropy term in the Gibbs Energy influences the equilibrium constant and calculation of reverse reaction rate constants with a large lever because of the temperature multiplier. Entropy also has a large contribution to preexponential factors. The advances in high level computational methods have resulted in accurate determination of thermochemical properties for linear and branched alkanes [
When determining the standard enthalpy of formation for a species, for example, it is common to implement different theoretical chemical equations that incorporate mass balance, with equivalent hybridization, and bond type on each side of a work equation. This allows for significant error cancellation via the use of a
For entropy and heat capacity calculations, the simple rigid-rotor harmonic-oscillator (HO) is utilized, almost exclusively, to describe the 3N-6 vibrations for nonlinear species and the corresponding contributions to entropy and heat capacities. It is well known that there are accuracy issues in determining the lower frequency torsions corresponding to internal rotations using this approximation. This can be corrected by replacing these frequencies with methods to treat the internal rotations as hindered rotors to increase accuracy for thermodynamic properties and reaction kinetics [
The initial research papers of Pitzer and Gwinn [
A number of studies utilizing different techniques for handling coupled and uncoupled internal rotor contributions have been reported [
The goal of the present study is to calculate entropy values for
Parent and radical C7H16 species in this study.
The parent and radical species were optimized at the B3LYP [
The three methods that we will use in our entropy analysis are given below.
Translations, vibrations, and external rotation (denoted TVR) contributions to entropy are determined using the Statistical Mechanics for Heat Capacity and Entropy (SMCPS) program [
This code is also used for the vibrations with corresponding torsion frequencies removed in the VIBIR and ROTATOR calculation for the internal rotor contributions below.
This method will analyze internal rotation contributions to entropy with the Pitzer and Gwinn [
VIBIR assumes that the rotational groups are symmetrical which is accurate for the primary and terminal methyl group rotation in these isomers. We also report results using VIBIR for nonsymmetrical internal rotors for comparison purposes.
The second method for internal rotation treatment is with the ROTATOR [
This method allows for a more accurate description of the internal rotor potential in the calculation of energy levels. Both symmetrical and unsymmetrical barriers can be accurately described for calculated contributions to entropies and heat capacities.
The calculation sets used to calculate entropy contributions in this study can be grouped into four methods. Abbreviations for these methods are denoted in italics below and used in Table
Comparison of calculated entropies for C7H16 parent species to available literature values.
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SMCPS ( |
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SMCPS/Methyl (18) | |
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SMCPS/VIBIR (18) | |
105.05 | 117.98 | 129.96 | 141.33 | 162.34 | 181.21 | 220.80 | SMCPS/ROTATOR (18) | |
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cc2cccc |
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SMCPS ( |
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SMCPS/Methyl (27) | |
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SMCPS/VIBIR (27) | |
103.70 | 116.75 | 128.87 | 140.37 | 161.54 | 180.50 | 220.20 | SMCPS/ROTATOR (27) | |
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cc2c2cc |
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SMCPS ( |
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SMCPS/Methyl (81) | |
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SMCPS/VIBIR (81) | |
99.44 | 112.53 | 124.65 | 136.13 | 157.27 | 176.20 | 215.84 | SMCPS/ROTATOR (81) | |
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Subtracted 0.026 cal mol−1 K−1 from values in Reference [
Comparison of our calculated entropies with the literature values [
Table
The data in Table
In a second analysis, each of the parent alkanes analyzed has the terminal methyl rotation contributions from the use of the Pitzer and Gwinn approximation method in the VIBIR code. These symmetrical terminal methyl rotations should be accurately calculated using this method. The contributions from the torsion frequencies associated with terminal methyl rotations are replaced with the VIBIR values and they are incorporated with the remaining vibration contributions at each temperature.
These SMCPS/Methyl values, in Figure
Calculation of entropy contributions using all of the internal rotors, which are considered to be uncoupled, was performed using both VIBIR and ROTATOR. Determining the corresponding rotation-vibration torsions using visualization in GaussView 5 was problematic due to the coupling of several internal rotors. To evaluate this, we tested several different vibration pair combinations where frequency evaluation in GaussView indicated torsions. It was observed that there was an insignificant change in the entropies using the various combinations. Internal rotations are typically characterized by lower vibrational frequencies as compared to other motions, so to maintain consistency the first six vibration frequencies for each species were replaced with the data from the respective rotor method. The data on these trials is included in the Supplementary Materials available online at
At 298 K, the SMCPS/VIBIR values provide accuracy within 1 cal mol−1 K−1 to the reference values for all of the parent species; SMCPS/ROTATOR provides a slightly closer value for cc2c2cc. As the temperature increases, the SMCPS/VIBIR values trend to underestimate the Scott values by up to 11–13 cal mol−1 K−1 at 1500 K. In contrast, SMCPS/ROTATOR provides a much closer approximation throughout the temperature range. This suggests that modeling all of the rotors, where the rotational barriers are similar or lower to those determined for these hydrocarbon parents and radicals results in a better approximation to entropy.
The HO approximation with corrections for internal rotations by the ROTATOR code provided the best comparison to the available literature reference values. It is important to note that SMCPS/VIBIR provides very good values at 298 K.
Entropies using the SMCPS/ROTATOR and SMCPS/VIBIR methods for the radical species are shown in Table
Comparison of calculated entropies for C7H16 radical species.
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cjcccccc (3) | 106.41 | 117.09 | 127.56 | 137.78 | 156.96 | 174.30 | 210.56 |
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ccjccccc (9) | 106.14 | 116.68 | 126.95 | 136.97 | 155.82 | 172.92 | 208.88 |
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cccjcccc (9) | 106.55 | 117.35 | 127.71 | 137.75 | 156.53 | 173.54 | 209.33 |
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ccccjccc (9) | 106.57 | 117.39 | 127.76 | 137.81 | 156.58 | 173.58 | 209.34 |
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cjc2cccc (9) | 105.12 | 116.02 | 126.60 | 136.86 | 156.03 | 173.31 | 209.45 |
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ccj2cccc (54) | 103.43 | 113.75 | 123.76 | 133.55 | 152.04 | 168.88 | 204.51 |
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cc2cjccc (27) | 103.94 | 114.91 | 125.37 | 135.44 | 154.20 | 171.13 | 206.77 |
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cc2ccjcc (27) | 102.28 | 113.40 | 124.04 | 134.28 | 153.30 | 170.42 | 206.31 |
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cc2cccjc (27) | 103.36 | 114.13 | 124.51 | 134.58 | 153.41 | 170.46 | 206.31 |
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cc2ccccj (9) | 103.60 | 114.48 | 125.04 | 135.28 | 154.44 | 171.71 | 207.88 |
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cjc2c2cc (27) | 102.45 | 113.64 | 124.28 | 134.53 | 153.59 | 170.79 | 206.91 |
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ccj2c2cc (81) | 102.41 | 113.00 | 123.25 | 133.19 | 151.82 | 168.72 | 204.36 |
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cc2cj2cc (81) | 102.01 | 112.63 | 122.80 | 132.62 | 151.00 | 167.67 | 202.97 |
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cc2c2jcc (27) | 101.47 | 112.59 | 123.24 | 133.50 | 152.62 | 169.86 | 206.01 |
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cc2c2cjc (81) | 100.32 | 111.32 | 121.90 | 132.10 | 151.07 | 168.18 | 204.10 |
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cc2c2ccj (27) | 101.71 | 112.87 | 123.53 | 133.78 | 152.84 | 170.03 | 206.08 |
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Values are from SMCPS/VIBIR and SMCPS/ROTATOR (in bold). Symmetry values are given in parenthesis for our methods.
SMCPS/VIBIR provides acceptable agreement with the SMCPS/ROTATOR values at 298 K as it did for the parent species. There is approximately a 2 cal mol−1 K−1 difference for cjc2cccc while the 2,3-dimethylpentane radicals all fall below a 1 cal mol−1 K−1 difference. This is not the case at higher temperatures, however, as the deviation increases with a minimum difference of 9 cal mol−1 K−1 at 1500 K between the two methods. VIBIR is based on the symmetrical barriers, while ROTATOR accurately models the complete potential energy curves for the internal rotation. In the SMCPS/VIBIR method all of the internal rotations which include asymmetrical nonmethyl rotations utilize the average barrier heights for the rotors. The error created by this approximation is not as pronounced at lower temperatures but is apparent in the larger deviation at increased temperatures. We attribute these differences to our use of the Pitzer and Gwinn method in our VIBIR code not being able to account for the nonsymmetrical internal rotor barriers, where the ROTATOR method does fully account for wells in nonsymmetrical barriers.
Since literature values are not available for the entropy of the radicals, our recommendation is to use the SMCPS/ROTATOR values in Table
Internal rotational barriers, moments of inertia, and vibrational frequencies were determined at the B3LYP/6-31G(d,p) level of theory for rotor contribution analysis. These potentials were used to determine entropies from 298 to 1500 K for
Entropies calculated using the SMCPS/ROTATOR method is recommended for the radical species. We plan on extending this analysis to include enthalpies, heat capacities, and bond dissociation energies in a future publication.