Chapter 24.2, Problem 17SSC

Interpretation Introduction

(a)

Interpretation:

The half life of a radioisotope is to be calculated if amount remains is 1.986 g from original mass of 2.00 g in two years.

Concept introduction:

The time needed for one half of nuclei of radioisotope to decay into its products is the half life. The formula to calculate remaining amount of radioactive element is,

$N=N_0{\left(\frac{1}{2}\right)}^n$........(1)

Where $N$ is remaining amount, $N_0$ is the initial amount and n is the number of half lives that have passed.

The n of equation (1) can be replaced with the equivalent quantity $t/T$ where t is the elapsed time and T is duration of half life. So, equation (1) becomes,

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

Answer to Problem 17SSC

Half life is 198 years.

Explanation of Solution

Given, Initial amount (N) is 2.00 g and amount remains $\left(N_0\right)$ is 1.986 g and elapsed time (t) is 2 years. Put all the values in equation (2).

$\begin{array}{c}N=N_0{\left(\frac{1}{2}\right)}^{t/T}\\ 1.986=2.00{\left(\frac{1}{2}\right)}^{\frac{2}{T}}\\ $ \frac{1.986}{2.00}={(0.5)}^{\frac{2}{T}}$$ 0.993={(0.5)}^{\frac{2}{T}}$\end{array}$

$\begin{array}{c}$ \ln 0.993=\ln {(0.5)}^{\frac{2}{T}}$\ln 0.993=\frac{2}{T}\ln (0.5)\\ T=\frac{2\ln (0.5)}{\ln 0.993}\\ =198\text{years}\end{array}$

(b)

Interpretation Introduction

Interpretation:

The amount of radioisotope remains after 10 years is to be calculated.

Concept introduction:

The time needed for one half of nuclei of radioisotope to decay into its products is the half life. The formula to calculate remaining amount of radioactive element is,

$N=N_0{\left(\frac{1}{2}\right)}^n$

Answer to Problem 17SSC

The amount of radioisotope remains after 10 years is 0.966 g.

Explanation of Solution

Elapsed time (t) is 10 years and half life (T) is 198 years. First calculate number of half lives.

$\begin{array}{c}\text{Half}\text{lives(}n\text{)=}\frac{t}{T}\\ =\frac{10}{198}\\ =0.050\end{array}$

The time needed for one half of nuclei of radioisotope to decay into its products is the half life. The formula to calculate remaining amount of radioactive element is,

$N=N_0{\left(\frac{1}{2}\right)}^n$

Now put the initial amount and half lives in the above equation.

$\begin{array}{c}N=N_0{\left(\frac{1}{2}\right)}^n\\ =2.00\text{g}{\left(\frac{1}{2}\right)}^{0.050}\\ =0.966\text{g}\end{array}$