# Iron-59 has a half-life of 45.1 days. How old is an iron nail if the Fe-59 content is 25% that of a new sample of iron? Show all calculations leading to a solution.

Aug 22, 2017

90 days

#### Explanation:

All radio decay follows 1st order kinetics and therefore is supported by the integrated rate law of a 1st order decay trend. The classic form of the 1st order decay equation is $A = {A}_{o} {e}^{-} \text{kt}$ where $A$ = final activity (or mass), ${A}_{o}$ = initial activity (or mass), $k$ = the rate constant & $t$ = time of decay. The rate constant $\left(k\right)$ as a function of half-life can be determined from $k \cdot {t}_{\frac{1}{2}} = 0.693$.

Given ${t}_{\frac{1}{2}} = 45.1 \mathrm{da} y s$ => $k = \left(\frac{0.693}{45.1}\right) \mathrm{da} y {s}^{-} 1$ = $0.0154 \mathrm{da} y {s}^{-} 1$

From the 1st order decay equation ...
$A = {A}_{o} {e}^{-} \text{kt}$ => $\ln \left(\frac{A}{A} _ o\right) = - k \cdot t$
=> $t = \left(\ln \frac{\frac{A}{A} _ o}{-} k\right)$ = $\left(\ln \frac{\frac{25}{100}}{-} 0.0154\right) \mathrm{da} y s$ = $90 \mathrm{da} y s$