Understanding the Concept:
The rate of a chemical or pharmacokinetic reaction is mathematically represented by the change in drug concentration ($C$) per unit time ($t$):
\[
\text{Rate} = -\frac{dC}{dt} = K_0 \cdot C^n
\]
Where $K_0$ is the rate constant and $n$ defines the order of the reaction. For a zero-order process, the rate of elimination is completely independent of the concentration of the drug remaining because the operating systems (such as metabolic enzymes or transport carriers) are fully saturated ($n = 0$).
Step 1: Set up the zero-order differential expression.
Substituting $n = 0$ into the rate equation:
\[
-\frac{dC}{dt} = K_0 \cdot C^0 \implies -\frac{dC}{dt} = K_0
\]
Step 2: Derive the units of the constant.
From the simplified relation, the units of the zero-order rate constant $K_0$ must match the units of the change in concentration over time:
\[
\text{Units of } K_0 = \frac{\text{Units of Concentration}}{\text{Units of Time}}
\]
Concentration is expressed as mass per unit volume (e.g., $\text{mg/ml}$), and time is expressed in hours ($\text{hour}$). Substituting these units gives:
\[
\text{Units of } K_0 = \frac{\text{mg/ml}}{\text{hour}} = \text{mg} \cdot \text{ml}^{-1} \cdot \text{hour}^{-1} = \text{mg/ml*hour}
\]