Question

The rate at which the concentration of a drug in the bloodstream decreases is proportional to the concentration at any time t. Initially, the concentration of the drug in the bloodstream is C0 g/mL. What is the concentration of the drug in the bloodstream any time t

Answer 1

Answer:

$C(t) = C_oe^{-kt}$

Explanation:

From the given information:

At any given time (t), let c(t) represent the concentration of the drug present in bloodstream.

Deriving the equation:

$\dfrac{dC}{dt}$ decrease proportionally to Concentration C

i.e

$\dfrac{dc}{dt} \alpha - C$

$\dfrac{dc}{dt} = -k C$

$\dfrac{dc}{c} = -k dt$

$\int \dfrac{dc}{c} = -k \int dt$

㏑(C) = -kt + λ

where,

λ is the integration constant.

Integrating at t = 0, concentration of blood = Co g/mL

C(0) = Co

㏑(C₀) = 0 + λ

λ = ㏑(C₀)

From ㏑(C) = -kt + λ

㏑(C) = -kt + ㏑C₀

㏑(C)  - ㏑C₀ =  -kt

$\dfrac{C}{C_o} = e^{-kt}$

$C(t) = C_oe^{-kt}$

∴

The concentration of drug in blood at any time t is:

$C(t) = C_oe^{-kt}$