Question

27–30. Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.

Valium metabolism The drug Valium is eliminated from the bloodstream with a half-life of 36 hr. Suppose a patient receives an initial dose of 20 mg of Valium at midnight. How much Valium is in the patient’s blood at noon the next day? When will the Valium concentration reach 10% of its initial level?

Accepted Answer

Identify the general form of the exponential decay function, which is given by $$C(t) = C_0$ \cdot e$$^{-kt}$$, where C(t$$)$$ is the concentration at time t$, C_0$ is the initial concentration, and k$ is the decay constant. Determine the reference point and units: here, t=0$ corresponds to midnight when the initial dose of 20 mg is administered, and time t$ is measured in hours. Use the half-life information to find the decay constant k$. Since the half-life T$$_{1/2} = 36$$ hours, apply the formula $$\frac{1}{2} = $$e^{-kT_{1/2}$$}$$, which can be rewritten as k$$ = \frac{\ln(2)}{T_{1/2}}$$. Calculate the amount of Valium in the bloodstream at noon the next day, which is 12 hours after midnight, by substituting t=12$ into the decay function: C(12) = 20$$ \cdot $$e^{-k \cdot 12}. To $$find when the concentration reaches 10% of the initial dose, set $$C(t) = 0.1 	imes 20$$ and solve for $$t$$ using the equation $$0.1 	imes 20 = 20 \cdot e$$^{-kt}$$, which simplifies to 0.1$$ = $$e^{-kt}. $$Taking the natural logarithm of both sides gives $$t = -\frac{\ln(0.1)}{k}.$$

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Concetti chiave

Exponential Decay Function

Half-Life and Decay Constant Relationship

Reference Point and Time Units