# Question e0dc9

Sep 11, 2017

The correct answer should be 624 hours (the data really only provides two significant figures).

#### Explanation:

This is a common nuclear chemistry problem. I’m not sure if it was correctly posted here.
The relationship between concentration and time for a first-order reaction (radioactive decay) is:
ln(([A]_t)/[A]_0)) = -kt where [A] is the concentration at time (t) and k is the rate constant.

Half-life is ${t}_{\frac{1}{2}} = \frac{0.693}{k}$
A 10% reduction means that ${\left[A\right]}_{t} / {\left[A\right]}_{0} = 0.9$, so $\ln \left(0.9\right) = - k \times 95$

$- 0.1054 = - k \times 95$ ; $k = \frac{0.1054}{95} = 0.00111$
Half-life is ${t}_{\frac{1}{2}} = \frac{0.693}{0.00111}$; ${t}_{\frac{1}{2}} = 624.3$ hours

CHECK:
ln(([A]_t)/[A]_0)) = -kt ; ln(([A]_624.3)/[A]_0)) = -0.00111 xx 624.3
IF it is the correct half-life, [A]_624.3)/[A]_0# = 0.5
$\ln \left(0.5\right) = - 0.693$ ; $- 0.693 = - 0.693$ CORRECT