What will the concentration be 224 minutes later?

Question

The rate at which a certain drug is eliminated by the body follows first-order kinetics, with a half life of 

 56 minutes.

Suppose in a particular patient the concentration of this drug in the bloodstream immediately after injection is 0.34 µg/mL. What will the concentration be 224 minutes later? Round your answer to 

 significant digits.

Answer 1

One way to answer this is to know that for a first order reaction FR = 0.5n where

FR = fraction remaining = ?

n = number of half lives that have elapsed = 224 min / 56 min = 4 half lives

FR = 0.5⁴ = 0.0625

Concentration remaining = 0.34 ug/ml x 0.0625 = 0.021 ug/ml (2 sig.figs.)

Another way is to use the integrated rate law for a 1st order reaction:

ln[A] = -kt + ln[A]₀ and for 1st order, t1/2 = 0.693/k and thus, k = 0.693/56 min = 0.012375 min^-1

ln[A] = -(0.012375)(224) + ln[0.34)

ln[A] = -2.772 + (-1.079) = -3.85

[A] = 0.021 ug/ml