Chemical interactions, heart arrhythmias, and morphogenesis processes are intimately associated with the development of spiral form and thereafter spiral instability. Periodic traveling waves are a possible feature of a dynamical system that can help explain spiral instability since from the core of regular spinning spirals, these kinds of waves can be released. This study deals with the persistence and stability analysis of periodic traveling waves in the diffusive chlorine dioxide-iodine-malonic acid model analytically and numerically. In the diffusive model, the emergence and uniqueness of the solution are examined. Additionally, we demonstrate that the diffusive model has a Hopf bifurcation, leading to periodic oscillating traveling waves. We calculate the stability threshold at which the periodic traveling waves become unstable due to the Eckhaus bifurcation. We investigate the spiral wave instability phenomena by using the numerical bifurcation analysis, where the spiral waves lead to far-field breakdown due to the Eckhaus instability of periodic traveling waves.