The specific rate constant of a reaction depends on the

Rate constant of a reaction depends only on temperature and catalyst.

$\mathrm{Rate}=\mathrm{K}{\left[\mathrm{C}\right]}^{\mathrm{n}}$

When concentration decreases rate also decreases but rate constant, 'K' does not changes

$\mathrm{Rate}=\mathrm{K}{\left[\mathrm{p}\right]}^{\mathrm{n}}$

When "pressure" decreases rate also decreases but rate constant constant, 'K' does not change.

As time passes by concentration changes occur and rate decreases but rate constant does not change.