We present a theory for describing nonclassical dynamics of reactions occurring in disordered media based on the fractional diffusion equation. An exact expression is derived for the Green's function required to calculate the survival probabilities of reactants. A novel temperature-dependent kinetic phase transition is found: The exponent gamma in the asymptotic power-law decay (proportional tot(-gamma)) of the geminate survival probability increases with temperature T below a critical temperature T-*, but decreases with T above T-*. The present theory explains in a unified manner the observed features of ligand-protein recombination reactions for a wide range of temperature and time scales.