Concept of Activation Energy in Unimolecular Reactions

An exact expression is derived, valid for all pressure regions, for the activation energy of a unimolecular reaction. Because of the reaction, the distribution of reactant molecules among vibrational levels deviates from a Boltzmann distribution; this deviation is explicitly taken into account. In the high‐pressure limit, the expression reduces to a well‐known form: $E_{\mathrm{act}}\mathrm{=〈EK〉\; /\; 〈K〉-〈E〉,}$ due originally to Tolman. In the low‐pressure limit our expression contains terms which do not appear in Tolman's formalism. These are a term E^M which depends on the deviation from a Boltzmann distribution, and a term E^Q which arises from the thermal average of energy‐dependent vibrational cross sections. It is shown that the low‐pressure expression for E_act does indeed yield an energy fairly close to the energy at which conversion from reactant to product occurs. An approximate analytic expression for E_act is obtained for the case that the cross sections for vibrational energy transfer can be approximated by a truncated harmonic oscillator model. A numerical study of the thermal decomposition of N₂O, using SSH theory to evaluate collision rates, shows that both E^M and E^Q make important contributions to the final value of E_act. It is also shown that the steady‐state approximation can lead to serious error in calculating low‐pressure rate constants and activation energies at high temperatures.