# Question #97af1

May 23, 2017

The half life is $= 126.6 s$

#### Explanation:

The equation of the radioactive decay is

$M \left(t\right) = {M}_{0} \cdot {e}^{- \lambda t}$

The initial mass is ${M}_{0} = 300 g$

The final mass is ${M}_{t} = 112 g$

The time is $t = 180 s$

The radioactive constant is $= \lambda$

Therefore,

$112 = 300 \cdot {e}^{- 180 \lambda}$

${e}^{- 180 \lambda} = \frac{112}{300} = 0.373$

${e}^{180 \lambda} = \frac{1}{0.373} = 2.679$

$180 \lambda = \ln \left(2.6790\right) = 0.985$

$\lambda = \frac{0.985}{180} = 0.0055$

But

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

So,

The half life is

${t}_{\frac{1}{2}} = \frac{0.693}{\lambda} = 126.6 s$