Minerals and organic matters in soils and sediments are chemically different and their ability to adsorb solute varies. Since individual minerals cannot be resolved in macroscopic models, these heterogeneous adsorption rates are often modelled as a random number with a pre-defined distribution function. Such an approach is also known as multiple-rate adsorption (MRA) models in reactive solute transport modelling. The MRA models have shown some improvement, but it is unclear that this improvement is due to their improved description of the statistics of the variable adsorption rates, or because of the extra parameters they introduced, which give them more freedom in curve-fitting. In this paper, we investigated this using pore-scale modelling. The pore-scale simulations were based on an idealised column packed with different minerals; solute concentration in the column was assumed to be low in comparison with the adsorptive capacity of the minerals. Hence, the adsorption of each mineral particle was assumed to be linear first-order kinetic. Both batch and column-displacement experiments at the pore scale were simulated using the lattice Boltzmann model. The simulated solute distributions were then volume-averaged to yield macroscopic concentration by treating the column as a continuum medium. The parameters in the MRA model were obtained from batch experiments and the dispersion coefficient was derived from displacement of an inert tracer. These parameters were then used to simulate the displacement of adsorptive solute under different water flow rates. The results showed that even though the MRA model accurately described the statistics of the heterogeneous adsorption rates, it failed to describe the macroscopic adsorption when the Pe number is high. We also found that the only way to improve the fitting is to increase the dispersion coefficient measured from the inert tracer. This yields a dispersion coefficient for adsorptive solute, which nonlinearly increases with the Pe number. To investigate whether this is a general phenomenon for adsorptive solute or only for solute under heterogeneous adsorption, we simulated solute displacement in the column filled by the same mineral. The results show that when the adsorption is homogeneous, the dispersion coefficients for adsorptive solute and inert tracer are the same. These findings imply that the MRA models should be used with care as the dispersion coefficient derived using inert tracers might not apply to adsorptive solute.