Growth and Decay Problem

Question

After the release of radioactive material into the atmosphere from a nuclear power plant in a country in 1984, the hay in that country was contaminated by a radioactive isotope (half-life 5 days). If it is safe to feed the hay to cows when 15% of the radioactive isotope remains, how long did the farmers need to wait to use this hay?

Answer 1

The definition of a half life is the amount of time it takes for the concentration of a reactant to decay to half of its initial value. For an exponential decay (like radioactive decay), this half life can be expressed as:

(eq 1)              ln([A]₀/[A]_t) = kt

The half life occurs (by definition) when [A]_t = [A]₀/2. Substituting this into our expression gives:

ln([A]₀/([A]₀/2)) = kt_1/2­, which simplifies to ln(2) = kt_1/2­, solving for t_1/2 gives:

(eq 2)              t_1/2 = 0.693/k, where k is a rate constant (rate of decay)

Because we know the half-life of the reaction, we can solve for the rate constant (eq 2). If we know how fast the material is decaying, then we can calculate how long it will take for only 15% of the material to remain by plugging the rate constant back into eq 1. We can start to get an idea from this graph (https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Nuclear_Chemistry/Nuclear_Kinetics/Half-Lives_and_Radioactive_Decay_Kinetics), which shows the relationship between concentration and time in a generic exponential decay function. Just eyeballing from here, we can see that it will take a little less than 3 half lives (15 days when t_1/2 = 5 days) for the radioactive material to reach 15% its original concentration.

Now that you have an idea of what you should get, do the math out and see if it agrees!

Hope this helps!

Answer 2

Fraction remaining = 0.5ⁿ where n is the number of half lives that have elapsed

0.15 = 0.5ⁿ and taking log of both sides, you have ...

-0.823 = -0.301n

n = 2.73 half lives

2.73 half lives x 5 days/half life = 13.7 days = 14 days