Non-universal dynamics is shown to occur in a one-dimensional non-equilibrium system of hard-core particles. The stochastic processes included are pair creation and annihilation (with rates e and e') and symmetric hopping rates which alternate from one bond to the next (Pa, Pb). A dynamical scaling relation between the relaxation time and the correlation length in the steady state is derived in a simple way for the case e' > Pa >> Pb >> e. We find that the dynamical exponent takes the non-universal value z = 2 ln(e'/e) / ln[(Pb e')/(Pa e)]. For the special condition e + e'= Pa + Pb, where the stochastic system is in principle soluble by reduction to a free fermion system, the model is mapped to the Glauber dynamics of an Ising chain with alternating ferromagnetic bonds of values J1 and J2, in contact with a quantum thermal bath. The full time dependence of the space-dependent magnetization and of the equal time spin-spin correlation function are studied by writing the master equation for this system in the quantum Hamiltonian formalism. In particular we obtain the dispersion relations and rigorously confirm the results obtained for the correlation length and for the dynamical exponent.Facultad de Ciencias Exactas