The rate of a reaction does depend on the concentrations of the reactants, as given by the rate law:
Rate = k[A]x[B]y
Where x and y are the orders with respect to each reactant.
However, the rate constant (k) is a proportionality factor that remains constant for a given reaction at a fixed temperature. Increases with temperature but is independent of the actual concentrations of the reactants.
Therefore, the rate constant is a characteristic of the reaction under given conditions and does not vary with changes in concentration.
