  A novel approach to the class of cosmic barotropic fluids in which the speed
of sound squared is defined as a function of the Equation of State parameter,
so called $c_s^2(w)$ models, is examined. For this class of models, a new
analytical reconstruction method is introduced for finding their equivalent
purely kinetic k-essence formulation. The method is explicitly demonstrated for
several $c_s^2(w)$ models. The application of the obtained explicit or closed
form solutions in understanding dark sector unification models is discussed.
