A symmetric discontinuous Galerkin scheme is applied for solving reactive transport problems. An explicit a posteriori error estimator in L²(L²) is derived for the semi-discrete scheme applied to transport with general kinetic reactions. Explicit a posteriori error estimators in terms of linear functionals and negative norms are also obtained for transport problems with simple reactions. An adaptive approach based on the a posteriori error estimators is proposed to modify the mesh dynamically. Numerical examples demonstrate the efficiency and effectivity of the estimators. Moreover, it is shown in this study that the proposed adaptivity eliminates the need of slope limiters.