An approximate analytic solution of the Blasius problem

The [4/3] Pade approximant for the derivative is modified so that the resulting expression has the required asymptotic behavior. This gives an analytical result which represents the solution of the classical Blasius problem on the whole domain.     

DFT investigation on the intramolecular and intermolecular proton transfer processes in 2-aminobenzothiazole (ABT) in the gas phase and in different solvents

DFT at the B3LYP/6-311++G(d,p) level of theory was performed to geometrically, thermodynamically, and kinetically investigate the tautomerism process of 2-aminobenzothiazole (ABT) with n water molecules (n = 1–3) and without water in the gas phase and in different solvents with a gradual increase in their dielectric constants. The geometries of the envisaged tautomers were optimized in the gas phase and in solution with the polarized continuum model (PCM). Equilibrium and rate constants for the forward and reverse intra-/intermolecular isolated and water-assisted tautomerism reactions were also calculated. The results suggest that the activation energy of the transition state of direct proton transfer in the isolated reaction is very high and that the rate constant is very slow (~ 10^?24 s), reflecting that the reaction is thermodynamically unfavored, whereas the barrier differences between the transition states of the tautomers decrease gradually as the number of water molecules increases from one to three. Moreover, the rate constants of the proposed reactions are ~ 10²³–10²⁵ faster than those of the isolated reaction, and the water-assisted tautomerism paths can be performed quickly, especially with the assistance of two molecules of water.