# Question #e70bd

Sep 11, 2017

This is what I get for parts (i) to (iii)

#### Explanation:

Nuclear decay formula using half-life is

${N}_{t} = {N}_{0} \times {2}^{\frac{- t}{T} _ \left(1 / 2\right)}$ ......(1)
where ${N}_{t}$ is sample of radioactive material remaining after time $t$, ${N}_{0}$ is initial amount of sample and ${T}_{1 / 2}$ is half life of the sample.

The formula can also be written as

${N}_{t} = {N}_{0} \times {e}^{\frac{- 0.693 t}{T} _ \left(1 / 2\right)}$ ......(2)

(i) Using equation (1)

${10}^{5} = {N}_{0} \times {2}^{\frac{- 32}{2}}$
$\implies {N}_{0} = {10}^{5} / {2}^{\frac{- 32}{2}}$
$\implies {N}_{0} = {10}^{5} \times {2}^{16}$
$\implies {N}_{0} = 6.5536 \times {10}^{9}$

(ii) Time $t$ when remaining sample A is equal to sample B can be calculated as the time when remaining sample ${N}_{t} = {N}_{0} / 2$, and sample B$= {N}_{0} / 2$.

We know that half-life for a given radioactive sample is the time for half the radioactive nuclei in that sample undergo radioactive decay.

$\therefore t = {T}_{1 / 2} = 2 s$

(iii) Inserting result of part (i) and given values in equation (1) we get

${N}_{t} = \left(6.5536 \times {10}^{9}\right) \times {2}^{\frac{- 4096}{2}}$
${N}_{t} = 390.625$
${N}_{t} = 390$, rounded to previous lower integer. (as fraction of nucleus can not decay.