Chapter 0.3, Problem 21E

a

To determine

Time taken to lose 50% of the sample.

Answer to Problem 21E

Time taken to lose 50% of the sample is 63 years.

Explanation of Solution

Given:

  $t_{1/2}=63years$

Concept Used:

The formula of half-life is:

  $N\left(t\right)=N_0{\left(\frac{1}{2}\right)}^{\frac{t}{t_{1/2}}}$

Where,

  $N\left(t\right)\to$ Remaining amount of substance

  $N_0\to$ Starting amount of substance

  $t\to$ Time elapsed

  $t_{1/2}\to$ Substance half-life

Calculation:

The amount reduces to 50% means $N\left(t\right)=\frac{1}{2}{N}_0$ .

Substitute the values in the formula:

  $\begin{array}{c}\frac{1}{2}{N}_0=N_0{\left(\frac{1}{2}\right)}^{\frac{t}{63}}\\ \frac{1}{2}={\left(\frac{1}{2}\right)}^{\frac{t}{63}}\end{array}$

Take logarithm with base $\frac{1}{2}$ to both sides:

  $\begin{array}{c}{\log }_{\frac{1}{2}}\left(\frac{1}{2}\right)={\log }_{\frac{1}{2}}\left[{\left(\frac{1}{2}\right)}^{\frac{t}{63}}\right]\\ {\log }_{\frac{1}{2}}\left(\frac{1}{2}\right)=\frac{t}{63}{\log }_{\frac{1}{2}}\left[\left(\frac{1}{2}\right)\right]\\ 1=\frac{t}{63}\left(1\right)\\ t=63years\end{array}$

Hence, time taken to lose 50% of the sample is 63 years.

b

To determine

Time taken to lose 75% of the sample.

Answer to Problem 21E

Time taken to lose 75% of the sample is 126 years.

Explanation of Solution

Given:

  $t_{1/2}=63years$

Concept Used:

The formula of half-life is:

  $N\left(t\right)=N_0{\left(\frac{1}{2}\right)}^{\frac{t}{t_{1/2}}}$

Where,

  $N\left(t\right)\to$ Remaining amount of substance

  $N_0\to$ Starting amount of substance

  $t\to$ Time elapsed

  $t_{1/2}\to$ Substance half-life

Calculation:

The amount reduces to 75% means $N\left(t\right)=\frac{1}{4}{N}_0$ .

Substitute the values in the formula:

  $\begin{array}{c}\frac{1}{4}{N}_0=N_0{\left(\frac{1}{2}\right)}^{\frac{t}{63}}\\ \frac{1}{4}={\left(\frac{1}{2}\right)}^{\frac{t}{63}}\end{array}$

Take logarithm with base $\frac{1}{2}$ to both sides:

  $\begin{array}{c}{\log }_{\frac{1}{2}}{\left(\frac{1}{4}\right)}^2={\log }_{\frac{1}{2}}\left[{\left(\frac{1}{2}\right)}^{\frac{t}{63}}\right]\\ 2{\log }_{\frac{1}{2}}\left(\frac{1}{2}\right)=\frac{t}{63}{\log }_{\frac{1}{2}}\left[\left(\frac{1}{2}\right)\right]\\ 2\left(1\right)=\frac{t}{63}\left(1\right)\\ t=126years\end{array}$

Hence, time taken to lose 75% of the sample is 126 years.