Question

For the following parallel chain reaction

what will be that value of overall half-life of $A$ in minutes? $\left[\text { Given that } \frac{[B]_{t}}{[C]_{t}}=\frac{16}{9}\right]$

Answer 1

We have,
$\frac{[B]_{t}}{[C]_{t}}=\frac{4 k_{1}}{3 k_{2}}=\frac{16}{9}$
so, $\frac{k_{1}}{k_{2}}=\frac{4}{3}$
Now, $k=k_{1}+k_{2}=\left[2 \times 10^{-3}+\frac{3}{4} \times 2 \times 10^{-3}\right] \sec ^{-1}$ $=\frac{7}{2} \times 10^{-3} sec ^{-1}=\frac{7 \times 10^{-3} \times 60}{2} \min ^{-1}$
so, $t_{1 / 2}=\frac{\ln 2}{7 \times 30 \times 10^{-3}} min$
$=\frac{693}{7 \times 30}=3.3\, \min$