Two solid samples are introduced into a constant medium oven and left for $1$ hour. Under these conditions, particles with an initial radius of $4$ mm undergo $58\%$ conversion, and particles of $2$ mm convert at $87\mathrm{.}5\%.$

a$)$ Find the mechanism controlling the reaction rate of this solid, considering that the size of the particles does not change over time.

b$)$ Find the time required for complete conversion of particles with a radius of $1$ mm$.$