Chapter 19, Problem 61A

Interpretation Introduction

Interpretation: The original mass of the radionuclide needs to be determined.

Concept introduction:  Radioactive decay follows the following first-order kinetics.

First order equation is given by,

  $t=\frac{1}{k}\text{ln}\frac{N_0}{N_t}\mathrm{.......}\text{(1) }$

  $\text{Where, }t=\text{ time required in radiactive decay }$

  $k=$ decay constant

  $N_0=$ initial amount of radionuclide

  $N_t=$ amount after time t

Relation between half-life and decay constant is $t_{1/2}=\frac{0.693}{k}$

So, from the equation (1) becomes, $t=\frac{t_{1/2}}{0.693}\text{ln}\frac{N_0}{N_t}$

Answer to Problem 61A

Mass of the original sample is $47.7g$ .

Explanation of Solution

Putting the appropriate values in final equation we get,

  $\begin{array}{c}140yr=\frac{35yr}{0.693}\text{ln}\frac{N_0}{3.0g}\\ \text{ln}\frac{N_0}{3.0g}=\frac{140\times 0.693}{35}\\ \text{ln}\frac{N_0}{3.0g}=2.77\\ \frac{N_0}{3.0g}=\text{exp(2}\text{.77)}\\ \frac{N_0}{3.0g}=15.9\\ N_0=(15.9\times 3.0)g\\ N_0=47.7g\end{array}$

Conclusion

Mass of the original sample is $47.7g$ .