Chapter 19, Problem 25A

Interpretation Introduction

Interpretation:The amount of each isotope of radium remaining after one month has to be calculated.

Concept introduction:Half-life can be defined as the time required for half of the starting material to decay. The amount of original isotope remaining after certain interval of time is calculated using the formula,

Amount of mass remaining= $\frac{1}{2^n}\left(originalmass\right)$

Answer to Problem 25A

The amount of $Ra$ remaining after one month is $\text{166}\text{.90}\text{mg}\text{.}$

The amount of $Ra$ remaining after one month is $\text{2}\text{.557}\text{mg}\text{.}$

The amount of $Ra$ remaining after one month is $\text{238}\text{.83}\text{mg}\text{.}$

Explanation of Solution

The time required for half of the starting material to decay is called half-life $\left(t_{1/2}\right).$ The amount of original isotope remaining after certain interval of time is calculated using the formula,

Amount of mass remaining= $\frac{1}{2^n}\left(originalmass\right)$

Where,

• $n$ represents the half-life.

The first three isotopes of radium are $Ra,$

  $Ra$ and $Ra.$

The half-life of $Ra$ is $\text{12}$ days.

The half-life of $Ra$ is $\text{3}\text{.6}$ days.

The half-life of $Ra$ is $15$ days.

The amount of $Ra$ remaining after one month is calculated is calculated as,

  $\begin{array}{l}n=\frac{\text{31}\text{days}}{\text{12}\text{days}}\\ n=\text{2}\text{.583}\end{array}$

Amount of sample remaining= $\frac{1}{2^n}\left(originalmass\right)$

Amount of sample remaining= $\frac{\text{1}}{{\text{2}}^{\text{2}\text{.583}}}\left(\text{1000}\text{mg}\right)$

Amount of sample remaining= $\text{166}\text{.90}\text{mg}$

The amount of $Ra$ remaining after one month is $\text{166}\text{.90}\text{mg}\text{.}$

The amount of $Ra$ remaining after one month is calculated is calculated as,

  $\begin{array}{l}n=\frac{\text{31}\text{days}}{3.6\text{days}}\\ n=\text{8}\text{.611}\end{array}$

Amount of sample remaining= $\frac{1}{2^n}\left(originalmass\right)$

Amount of sample remaining= $\frac{\text{1}}{{\text{2}}^{8.611}}\left(\text{1000}\text{mg}\right)$

Amount of sample remaining= $\text{2}\text{.557}\text{mg}$

The amount of $Ra$ remaining after one month is $\text{2}\text{.557}\text{mg}\text{.}$

The amount of $Ra$ remaining after one month is calculated is calculated as,

  $\begin{array}{l}n=\frac{\text{31}\text{days}}{15\text{days}}\\ n=\text{2}\text{.066}\end{array}$

Amount of sample remaining= $\frac{1}{2^n}\left(originalmass\right)$

Amount of sample remaining= $\frac{\text{1}}{2^{2.066}}\left(\text{1000}\text{mg}\right)$

Amount of sample remaining= $\text{238}\text{.83}\text{mg}$

The amount of $Ra$ remaining after one month is $\text{283}\text{.83}\text{mg}\text{.}$

Conclusion

The amount of $Ra$ remaining after one month is $\text{166}\text{.90}\text{mg}\text{.}$

The amount of $Ra$ remaining after one month is $\text{2}\text{.557}\text{mg}\text{.}$

The amount of $Ra$ remaining after one month is $\text{238}\text{.83}\text{mg}\text{.}$