In a one-compartment model, which equation correctly expresses the elimination rate constant k in terms of clearance (CL) and volume of distribution (Vd)?

• A. k = Dose / Vd
• B. k = Vd / CL
• C. k = F × CL / Vd
• D. k = CL / Vd

Answer: (D)

In a one-compartment model with first-order elimination, the rate constant k is determined by how quickly the drug is cleared relative to how much volume the drug is distributed in. The body eliminates drug at a rate = clearance × concentration, and the concentration is the amount in the body divided by the volume of distribution: C = A / Vd. The rate of change of the drug amount is dA/dt = -k A, but it can also be written as dA/dt = - (CL × C) = - (CL × A / Vd). Equating these gives k = CL / Vd. This makes the units consistent: clearance (volume/time) divided by volume (volume) yields 1/time, which is the unit of k. It also leads to the practical relation for half-life, t1/2 = 0.693 / k = 0.693 × Vd / CL.

Notes: for non-IV administration, you use apparent CL and Vd (CL/F and Vd/F); the ratio CL/Vd remains the same because F cancels in the derivation. The other forms would mix in incorrect units or unnecessary factors, so they do not represent the elimination rate constant.