Units of rate of first order reaction are 

$\mathrm{rate}=\frac{\mathrm{dc}}{\mathrm{dt}}=\frac{\mathrm{mole}.{\mathrm{lit}}^{-1}}{\mathrm{time}}=\mathrm{mole}.{\mathrm{lit}}^{-1}.{\mathrm{time}}^{-1}$

Units of rate do not vary with order of the reaction. Units of rate constant depend on order of the reaction.