Step 1: Understand the rate constant.
For a first order reaction, the rate constant (k) does not depend on the concentration of reactants or products but on the time for the reaction to occur.
Step 2: Analyze the options.
- Concentration of reactants (Option A) affects the rate of reaction but not the rate constant.
- Concentration of products (Option B) is similarly related to reaction rate, not the rate constant.
- Time (Option C) is the correct factor that the rate constant of a first order reaction depends on, since the rate constant remains constant over time.
- Temperature (Option D) does affect the rate of reaction but not the rate constant for a given reaction.
Step 3: Apply to the options.
The correct answer is (C) time.
Final Answer:
\[
\boxed{\text{time}}
\]