Chapter 8.8, Problem 11PPS

To determine

To find: The half life of substance .

Answer to Problem 11PPS

The half life is $t=20.38$ year.

Explanation of Solution

Given information:

The given function is substance decays $99.9\%$ total mass after $200$ year.

Calculation:

Calculate the equation.

  $\begin{array}{c}y=a{e}^{-kt}\\ \frac{0.1}{100}=e^{-k200}\\ 0.001=e^{-k200}\\ \ln 0.001=\ln e^{-k200}\\ -6.9=200k\\ k=0.034\\ \end{array}$

Substitute the value of decay equation.

  $\begin{array}{c}y=a{e}^{-0.034t}\\ 0.5a=a{e}^{-0.34t}\\ 0.5=e^{-0.034t}\\ \ln 0.5=-0.034t\\ t=20.38\end{array}$

Therefore the half life is $t=20.38$ year.