Chapter 7.4, Problem 21E

To determine

To Find: The half-life of $\text{Sm-151}$

Answer to Problem 21E

half-life of $\text{Sm-151}$ $\approx 90.01years$

Explanation of Solution

Given information: The radioactive decay of $\text{Sm-151}$ can be modelled by the differential equation $\frac{dy}{dt}=-0.0077y$ , where $t$ is measured in years

Concept used: If $y$ is the mass remaining from an initial mass $y_0$ of the substance after time $t$ , then the mass decays exponentially

  $y=y_0{e}^{-kt}$

Where $k$ is a constant

Half-life is defined as the time it takes for one-half of the atoms of a radioactive material to disintegrate

Calculation: the differential equation $\frac{dy}{dt}=-0.0077y$

Dividing by $y$ both sides

  $\frac{1}{y}\frac{dy}{dt}=-0.0077$

Multiply by $dt$ both sides

  $\frac{dy}{y}=-0.0077dt$

Initial mass $=y_0$

After time $t$ the mass remaining from an initial mass $=y$

Integrate both sides

  $\begin{array}{l}{\displaystyle \int \frac{dy}{y}=}{\displaystyle \int -0.0077dt}\\ \ln y=-0.0077t+c\end{array}$

When $t=0$

the initial mass is $y_0$

  $\ln y_0=c$

Substituting value of $c$ into equation $\ln y=-0.0077t+c$

  $\begin{array}{l}\ln y=-0.0077t+\ln y_0\\ \ln y-\ln y_0=-0.0077t\\ \ln \frac{y}{y_0}=-0.0077t\\ \frac{y}{y_0}=e^{-0.0077t}\\ y=y_0{e}^{-0.0077t}\end{array}$

Half-life is defined as the time it takes for one-half of the atoms of a radioactive material to disintegrate

Let Half-life $=t$

And $y=\frac{y_0}{2}$

Substituting value of Half-life and $y$ into equation $y=y_0{e}^{-0.0077t}$

  $\begin{array}{l}\frac{y_0}{2}=y_0{e}^{-0.0077t}\\ \frac{1}{2}=e^{-0.0077t}\\ t\approx 90.01year\end{array}$

half-life of $\text{Sm-151}$ $\approx 90.01years$