Step 1: Recall what a rate constant represents.
The rate constant $k$ relates the speed of a chemical reaction to the concentrations of the reacting species.
Its dimensional units are not fixed and depend on the overall order of the reaction.
Step 2: Write the general expression for units.
For a reaction of order $n$, the units of the rate constant are given by:
\[ (\text{concentration})^{\,1-n} \times (\text{time})^{-1} \]
Since concentration is usually expressed in $\text{mol L}^{-1}$, this form can be applied directly.
Step 3: Apply the formula to a first-order reaction.
For a first-order reaction, $n = 1$.
Substituting into the general expression:
\[ (\text{mol L}^{-1})^{\,1-1} \times s^{-1} \]
\[ (\text{mol L}^{-1})^{0} \times s^{-1} = s^{-1} \]
This shows that the rate constant for a first-order reaction does not depend on concentration units.
Step 4: Final conclusion.
The unit of the rate constant for a first-order reaction is:
\[ \boxed{s^{-1}} \]