Chapter 6.3, Problem 17P

A radioactive substance $R_1$ having decay rate $k_1$ disintegrates into a second radioactive substance $R_2$ having decay rate $k_2$. Substance $R_2$ disintegrates into $R_3$, which is stable. If $m_i\left(t\right)$ represents the mass of substance $R_i$ at time $t$, $i=1,2,3,$ the applicable equations are

$\begin{array}{ccc}m{\text{'}}_1=& -k_1{m}_1& \\ m{\text{'}}_2=& k_1{m}_1-& k_2{m}_2\\ m{\text{'}}_3=& k_2{m}_2& \end{array}$

Draw a block diagram of a compartment model of the overall reaction. Label the directed arrows that represent the mass flows between compartments with the appropriate radioactive decay rate constants. Use the eigenvalue method to solve the above system under the conditions

$m_1\left(0\right)=m_0,m_2\left(0\right)=0,m_3\left(0\right)=0$