Assessing solute transport in rivers is crucial for water quality management and environmental protection. This paper presents a comprehensive study on one-dimensional solute transport that incorporates advection, rate-limited adsorption on suspended sediments, first-order degradation, and solute accumulation in the riverbed, modeled through deposition and erosion processes. Analytical solutions are derived using the Laplace transform method in a hypothetical case with no erosion or deposition. These solutions are then used to evaluate the numerical solutions obtained using a first-order explicit finite difference scheme (upwind). Two scenarios are considered, instantaneous discharge, involving an initial condition, and continuous discharge, involving a boundary condition. Furthermore, the accumulation of solute in the bed is investigated for both discharge scenarios in an average case where erosion and deposition are in equilibrium. This study offers insights into understanding the solute behavior and the influence of various transport and chemical processes and their impact on solute fate and transport.