The goal of this article is to formulate and analyze the simplest logistic pair-formation model and to contrast its dynamics to that of the corresponding Malthusian pair-formation model, that is, the Kendall/Keyfitz and Dietz/Hadeler model. The Malthusian pairformation model supports a unique nontrivial stable exponential solution and we show that the logistic pair-formation model also supports a unique stable nontrivial bounded solution.
Adopting BYLP method in Density Functional Theory (DFT), we make theoretical study on the ozonide-orthophenylphenol, parachlorophenol, orthobenzoquinone and parabenzoquinone in the two reaction routes of phenol oxidizing into benzoquinone with ozone. We get the geometric configuration of molecules, charge distribution of atoms, thermodynamical properties and frontier orbit energy. Natural Bond Orbital(NBO)charge calculation shows that compared with orthobenzoquinone and parabenzoquinone molecules, phenol, orthophenylphenol and parachlorophenol molecules have stronger reactivity and they are more likely to have electrophilic substitution reaction. Thermodynamic properties indicate that phenol is easy to have oxidation reaction and produce orthophenylphenol which is easily to oxidize into orthobenzoquinone no matter at low temperature, room temperature or high temperature. Another reaction pathwaycalculation shows that in thermodynamics, phenol will not easily ozonize into parachlorophenol; while parachlorophenol will easily ozonize into parabenzoquinone. Frontier orbit energy calculation shows that phenol, orthophenylphenol, and parachlorophenol show similar stability. Orthobenzoquinone and parabenzoquinone have the strongest stability.
Using DFT/6-31+G (d, p) method, the structure of phenol are gained in the global optimization and properties were theoretically studied. The atomic electric charges, activation of reaction and thermodynamics parameters are obtained. The calculation shows that benzene ring in phenol tends to have electrophonic attacking substitution reaction O3 directly and form catechol and hydroquinol. The calculation of thermodynamics properties indicate that two pathways are exothermic reactions, and the Gibbs free energies (ΔG) are always less than zero, two reactions are easily occurred spontaneously. Dynamics calculations show that there is only one transition state in each reaction; through vibrational analysis we confirm the transition state. After corrected single point energy, we find that the reaction activation energies of the two reactions are small (Ea1=4.48kcal/mol and Ea2=2.87kal/mol), indicating that ortho-position and para-position products exist simultaneously, which is in accordance with the thermodynamics calculation result.
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