After an aqueous solution containing Sr2+ was flowed through a bed packed with particles of activated charcoal for a very long time the flow was abruptly switched to deionized water. The variation in ionic concentration at the outlet with time was characterized by the power law C(out)similar to t-(alpha), for sufficiently large t. This fractal-like behavior we reported previously has further been studied theoretically and experimentally. A major concern is to examine dependency of alpha on experimental conditions. Adsorption sites predominate over dean-end pores or spaces as "trappers" in our case where Sr2+ or Ba2+ is adsorbed on activated charcoal. The experimental results can well be explained by our theoretical model. It has been found, however that the response curve for Sr2+ deviates from the power law after a very long time. This deviation can be explained by introducing the assumption that there exists a maximum value of the activation energy for the desorption process. The curve for Ba2+, on the other hand, exhibits no such deviation until C-out decreases to the detection limit of the analytical device used.