Chapter 7, Problem 12RE

To determine

To find : the mass that remains from a $100mg$ sample after $20$ years

Answer to Problem 12RE

The mass of the sample is $7.10mg$

Explanation of Solution

Given information : The question is

Cobalt- $60$ has a half life of $5.24$ years

Calculation :

If $h$ is the half life and $m_0$ is the initial mass of a substance, then the mass remain t years

  $m\left(t\right)=m_0{e}^{kt}$

Where $k=\frac{\ln \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{h}$

  $\begin{array}{l}k=\frac{\ln \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{5.24}\approx -1.3228\\ \Rightarrow m\left(t\right)=100e^{-1.3228t}\\ \Rightarrow m\left(20\right)=100e^{-1.3228\left(20\right)}\approx 7.10mg\end{array}$

b.

To determine

How long would it take for the mass to decay to $1mg$

Answer to Problem 12RE

It takes $34.8years$ for the mass to decay to $1mg$

Explanation of Solution

Given information : Given information : The question is

Cobalt- $60$ has a half life of $5.24$ years

Calculation :

If $h$ is the half life and $m_0$ is the initial mass of a substance, then the mass remain t years

  $m\left(t\right)=m_0{e}^{kt}$

Where $k=\frac{\ln \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{h}$

  $\begin{array}{l}k=\frac{\ln \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{5.24}\approx -1.3228\\ \Rightarrow m\left(t\right)=100e^{-1.3228t}\\ \Rightarrow 1=100e^{-1.3228t}\\ \Rightarrow 0.1=e^{-1.3228t}\\ \Rightarrow \ln 0.1=-1.3228t\\ \Rightarrow t=\frac{\ln 0.1}{-1.3228}\approx 34.8years\end{array}$