Question

Beryllium-li decomposes into boron-li with a half-life of

13.8 seconds. How long will it take 2400 g of beryllium-il
to decompose into 75 g of beryllium-il? (3 marks)

Answer 1

`\textit{Amount for Exponential Decay using Half-Life} \\\\ A=P\left( (1)/(2) \right)^{(t)/(h)}\qquad \begin{cases} A=\textit{current amount}\dotfill &75\\ P=\textit{initial amount}\dotfill &2400\\ t=seconds\\ h=\textit{half-life}\dotfill &13.8 \end{cases} \\\\\\ 75=2400\left( (1)/(2) \right)^{(t)/(13.8)}\implies \cfrac{75}{2400}=\left( (1)/(2) \right)^{(t)/(13.8)}\implies \cfrac{1}{32}=\left( (1)/(2) \right)^{(1)/(13.8) t}`

`\log\left( \cfrac{1}{32} \right)=\log\left[ \left( (1)/(2) \right)^{(1)/(13.8) t} \right]\implies \log\left( \cfrac{1}{32} \right)=t\log\left[ \left( (1)/(2) \right)^{(1)/(13.8)} \right] \\\\\\ \cfrac{\log\left( (1)/(32) \right)}{\log\left[ \left( (1)/(2) \right)^{(1)/(13.8)} \right]}=t\implies \cfrac{\log\left( (1)/(32) \right)}{\log\left[ \sqrt[13.8]{(1)/(2)} \ \right]}=t\implies 69\approx t`