Chapter 19.1, Problem 7RQ

Interpretation Introduction

Interpretation: Given its half life, time taken by a $1.000$ mol radioactive sample to reach $0.125$ mol is to be given and explained.

Concept introduction: Half-life of a radioactive sample is the time taken by it to reduce its concentration to half. Formula used to calculate the amount of sample left after half-lives is

  $\text{Half-life}=\frac{1}{2^n}$

Answer to Problem 7RQ

Time taken by a $1.000$ mol radioactive sample to reach $0.125$mol is

Explanation of Solution

Half-life of the givenradioactive sample $=6hrs$

Formula used to calculate the amount of sample left after half-lives is

  $\text{Half-life}=\frac{1}{2^n}$ ... (1)

Fraction of radioactive sample left $=0.125$

  $=\frac{1}{8}$ ... (2)

Equating (1) and (2),

  $\begin{array}{l}\frac{1}{8}=\frac{1}{{\left(2\right)}^n}\\ \frac{1}{{\left(2\right)}^3}=\frac{1}{{\left(2\right)}^n}\end{array}$

  $n=3$

So it took 3 half-lives for the given radioactive sample of $1.000$ mol to reach $0.125$ mol.

Time taken $\begin{array}{l}=3\times 6hrs\\ =18hrs\end{array}$

Conclusion

Time taken by a $1.000$ mol radioactive sample to reach $0.125$mol is