# The half-life of cobalt-60 is 5.27 years. Approximately how much of a 199 g sample will remain after 20 years?

Apr 18, 2016

$14.3 \text{g}$

#### Explanation:

The expression for radioactive decay is:

${N}_{t} = {N}_{0} {e}^{- \lambda t}$

${N}_{0}$ is the initial number of undecayed atoms.

${N}_{t}$ is the number of undecayed atoms remaining at time $t$

$\lambda$ is the decay constant

The relationship between $\lambda$ and the half - life ${t}_{\frac{1}{2}}$ is:

$\lambda = \frac{0.693}{t} _ \left(\frac{1}{2}\right)$

$\therefore \lambda = \frac{0.693}{5.27} = 0.1315 {\text{a}}^{- 1}$

Taking natural logs of both sides of the decay expression $\Rightarrow$

$\ln {N}_{t} = \ln {N}_{0} - \lambda t$

$\therefore \ln {N}_{t} = \ln 199 - \left(0.1315 \times 20\right)$

$\ln {N}_{t} = 5.293 - 2.63 = 2.66$

From which:

${N}_{t} = 14.29 \text{g}$