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{\bf Question}

Consider a sequential reaction

$$A\stackrel{k_1}{\longrightarrow}B\stackrel{k_2}{\longrightarrow}C$$

in which atoms of type $A$ decay into atoms of type $B$, and atoms
of type $B$ decay into atoms of type $C$, which are stable. Let
$a(t),b(t),c(t)$ be the number of atoms at time $t$ of type
$A,B,C$ respectively. The numbers of atoms are large, so that they
may taken to vary continuously, and the processes may then be
modelled by a pair of coupled first order differential equations
describing the rates of the decay:

$$\frac{da}{dt}=-k_1 a,\ \frac{db}{dt}=k_1 a-k_2 b,$$

together with the information that $a(t) + b(t) + c(t)$ is
constant. Assuming that at time $t=0$, $a(0)=\alpha$ and
$b(0)=0=c(0)$ find expressions for the number of atoms of each
type which are present at any given time.



{\bf Answer}

Number of atoms of type A:

Solve $\displaystyle \frac{\partial a}{\partial t}=-{K_1}a$
(Separation of variables)

$\displaystyle a(t)=\lambda e^{-K_1t} (\lambda\ \rm constant)$

Using boundary condition, $a(0)=\alpha$ gives $\lambda = \alpha$

so $\displaystyle \underline {a(t)=\alpha e^{{-K_1}t}}$

\bigskip
Number of atoms of type B:

Solve $\displaystyle \frac{\partial b}{\partial t} = K_1 a - K_2 b
= K_1 \alpha e^{{-K_1}t} - K_2 b$ (from above)

This gives a first order linear differential equation

$$\frac {\partial b}{\partial t}+K_2 b = K_1 \alpha e^{-K_1 t}$$

with integrating factor: $\displaystyle I(t)=e^{K_2 \int \,dt} =
e^{K_2 t}$

Multiplying equation by integrating factor gives

$$e^{K_2 t} \frac{\partial b}{\partial t} + K_2 b e^{K_2 t} = K_1
\alpha e^{(K_2-K_1)t}$$

so

$$\frac{\partial{}}{\partial t}(b e^{K_2 t})=K_1 \alpha
e^{(K_2-K_1)t}.$$

Integrating gives

$$b e^{K_2 t}=\frac{K_1 \alpha}{K_2-K_1}e^{(K_2-K_1)t}+D$$

where D is constant.

Boundary conditions $b(0)=0$ gives

$$0=\frac{K_1 \alpha}{K_2-K_1}+D \Rightarrow D=-\frac{K_1
\alpha}{K_2-K_1}$$

Solution is

$$\underline {b(t)=\frac{K_1 \alpha}{K_2-K_1} \left (e^{-K_1
t}-e^{-K_2 t} \right)}$$

\bigskip

Number of atoms of type C: $c(t)=K-a(t)-b(t)$ for some constant
$K$. Using boundary conditions $a(0)=\alpha,\ b(0)=0=c(0)$ we have

$$0=K-\alpha-0 \Rightarrow K=\alpha$$

so
\begin{eqnarray*}
c(t) & = & \alpha - \alpha e^{-K_1 t} - \frac{K_1 \alpha}{K_2 -
K_1} \left( e^{-K_1 t} - e^{-K_2 t} \right) \\ & = & \underline {
\alpha \left ( 1 + \frac{K_2}{K_1 - K_2} e^{K_1 t} -
\frac{K_1}{K_1-K_2} e^{-K_2 t} \right )}
\end{eqnarray*}


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