Effect of energy dependence of the density of states on pressure-dependent rate constants
The FE integral for the normalized Boltzmann-weighted number of unimolecular states above the threshold energy is the key quantity for computing the collision efficiency in the pressure-dependent unimolecular rate theory developed by Troe, who calls this the energy dependence factor of the density of states. By using the Whitten–Rabinovitch approximation and assuming that the Whitten–Rabinovitch a(E) function is independent of energy, FE can be approximated by an analytical formula; this approximate formula is widely used because of its convenience and computational efficiency. Here we test its validity by comparing the rate constants computed by using the approximate FE to the ones determined by using the numerically integrated FE. For small-sized molecules and for reactions with high threshold energies E0, the differences are negligible at all temperatures, but in other cases, the approximate formula tends to underestimate FE and thus overestimates the collision efficiency, and this leads to smaller pressure falloff. When a(E) at high energies differs appreciably from a(E0), we find that the underestimation of pressure-dependent rate constants by using the approximate formula can be greater than a factor of 5 at high temperatures. The physical insight we draw from this study is that, for reactions with threshold energies below about 30 kcal mol−1, the rate of collisional energy transfer can be appreciably slowed down by the increase in the density of states at higher energies, and this increases the falloff effect by which finite-pressure rate constants become lower than the high-pressure limit, especially at higher temperatures.