  The Cattaneo-Vernotte model has been widely studied to take momentum
relaxation into account in transport equations. Yet, the effect of reactions on
the Cattaneo-Vernotte model has not been fully elucidated. At present, it is
unclear how the current density associated with reactions can be expressed in
the Cattaneo-Vernotte model. Herein, we derive a modified Cattaneo-Vernotte
model by applying the projection operator method to the Fokker-Planck-Kramers
equation with a reaction sink. The same modified Cattaneo-Vernotte model can be
derived by a Grad procedure. We show that the inertial effect influences the
reaction rate coefficient differently depending on whether the intrinsic
reaction rate constant in the reaction sink term depends on the solute relative
velocity or not. The momentum relaxation effect can be expressed by a modified
Smoluchowski equation including a memory kernel using the Cattaneo-Vernotte
model. When the intrinsic reaction rate constant is independent of the reactant
velocity and is localized, the modified Smoluchowski equation should be
generalized to include a reaction term without a memory kernel. When the
intrinsic reaction rate constant depends on the relative velocity of reactants,
an additional reaction term with a memory kernel is required because of
competition between the current density associated with the reaction and the
diffusive flux during momentum relaxation. The competition effect influences
even the long-time reaction rate coefficient.
