The one-dimensional convection-dispersion model is investigated with respect to the proper boundary conditions to impose in applications to tracer movement in freely draining soil columns. It is shown that the model can be interpreted probabilistically and that this interpretation leads to physical restrictions on solute molecule transport through the inlet and exit boundaries of a soil column. These constraints, in turn, show that the correct exit boundary condition for a freely draining soil column is zero resident concentration or, equivalently, zero gradient in the flux concentration. A new solution is presented for the resident concentration under a closed inlet boundary.