4g of a radioactive material of half-life 10 days is spilled on a laboratory floor. How long would it take to disintegrate \(3.5\) g of material?
Explanation
Given; \(t_{k}=10\) days,
Original mass \(N _{0}=4 g\)
Mass disintegrated \(=3.5 g\)inal mass or mass remaining,
\(N =4-3.5=0.5 g\)
Using the formula;
$$
\begin{aligned}
\left(\frac{N}{N_{0}}\right) &=\left(\frac{1}{2}\right)^{\frac{T}{1 / 2}} \\
\frac{0.5}{4} &=\left(\frac{1}{2}\right)^{\frac{T}{10}} \\125 &=(0.5)^{\frac{T}{10}} \\
\log 0.125 &=\frac{T}{10} \log 0.5 \\
T &=\frac{10 \log 0.125}{\log 0.5}=10 \times 3 \\
&=30 \text { days. }
\end{aligned}
$$