In this paper, a reaction-diffusion Sporn-Seelig model subject to homogeneous Neumann boundary condition in the one dimensional spatial open bounded domain is considered. Of our particular interests, we are concerned with diffusion-driven instability of both the positive constant equilibrium solution and the Hopf bifurcating spatially homogeneous periodic solutions. To strengthen our analytical results, we also include some numerical simulations. These results allow for the clearer understanding the mechanisms of the spatiotemporal pattern formations of this chemical reaction model.