Chapter 19, Problem 8E

(a)

To determine

The number of atoms of radioactive substance with half-life 12 minutes and initial number of atoms of $102,400$ remain after 48 minutes

Answer to Problem 8E

The number of atoms of the element after 48 minutes are $6400\text{atoms}$.

Explanation of Solution

Given info: The initial number of atoms are $102,400$ and half-life of the element is $12\text{minutes}$.

Half-life of a radioactive element is the time taken by the element to decay into half of its initial amount. It is represented by $t_{\frac{1}{2}}$.

Let n be the number of half-life involved in the decay of a radioactive element.

Write the expression for the number half-life.

$\frac{t}{t_{\frac{1}{2}}}=n$

Here,

$t$ is the time

$t_{\frac{1}{2}}$ is the half-life of the radioactive element

$n$ is the number half-life

Substitute $48\text{minutes}$ for $t$ and $12\text{minutes}$ for $t_{\frac{1}{2}}$ in the above equation to get $n$.

$\begin{array}{c}n=\frac{48\text{minutes}}{12\text{minutes}}\\ =4\end{array}$

In each half life, half portion of amount decays. Therefore write the fraction of initial element remaining after 4 half-life.

$\begin{array}{c}\text{fraction}\text{remaining}\text{=}\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\\ =\frac{1}{16}\end{array}$

Therefore, write the expression for the number of atoms decayed after $4\text{minutes}$.

$N\text{=}{N}_0\times \text{fraction}\text{remaining}$

Here,

$N$ is the number of atoms remaining

$N_0$ is the number of atoms present at initial time

Substitute $102,400\text{atoms}$ for $N_0$ and $\frac{1}{16}$ for fraction in the above equation to get $N$.

$\begin{array}{c}N=102,400\text{atoms }\times \frac{1}{16}\\ =6400\text{atoms}\end{array}$

Conclusion:

The number of atoms of the element after 48 minutes are $6400$.

(b)

To determine

The number of atoms of radioactive substance with half-life 12 minutes and initial number of atoms of $102,400$ remain after I hour 24 minutes

Answer to Problem 8E

The number of atoms of the atoms remain after 1 hour 24 minutes $800\text{atoms}$.

Explanation of Solution

Given info: The initial number of atoms are $102,400$ and half-life of the element is $12\text{minutes}$.

Write the expression for the number half-life.

$\frac{t}{t_{\frac{1}{2}}}=n$

Convert 1 hour 24 minutes into minutes.

$\begin{array}{c}\text{ I hour 24 minutes}\text{=1}\text{hour}\times \frac{60\text{minutes}}{1\text{hour}}+24\text{minutes}\\ \text{=}\text{84}\text{minutes}\end{array}$

Substitute $\text{84}\text{minutes}$ for $t$ and $12\text{minutes}$ for $t_{\frac{1}{2}}$ in the above equation to get $n$.

$\begin{array}{c}n=\frac{84\text{minutes}}{12\text{minutes}}\\ =7\end{array}$

In each half life, half portion of amount decays. Therefore write the fraction of initial element remaining after 7 half-life.

$\begin{array}{c}\text{fraction}\text{remaining}\text{=}\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\\ =\frac{1}{128}\end{array}$

Therefore, write the expression for the number of atoms decayed after $84\text{minutes}$.

$N\text{=}{N}_0\times \text{fraction}\text{remaining}$

Substitute $102,400\text{atoms}$ for $N_0$ and $\frac{1}{128}$ for fraction in the above equation to get $N$.

$\begin{array}{c}N=102,400\text{atoms }\times \frac{1}{128}\\ =800\text{atoms}\end{array}$

Conclusion:

The number of atoms of the atoms remain after 1 hour 24 minutes $800\text{atoms}$.