We consider a model of the reaction on the integer lattice in which particles do not move while particles move as independent continuous time, simple symmetric random walks. particles are transformed instantaneously to particles upon contact. We start with a fixed number of particles at each site to the right of the origin. We prove a central limit theorem for the rightmost visited site of the particles up to time and show that the law of the environment as seen from the front converges to a unique invariant measure.