Question

$A\stackrel{k_A}{\underset{T_A}{⟶}}$ product, $B\stackrel{k_B}{\underset{T_B}{⟶}}$ product
A and B are two radioactive elements with half-life periods $T_A$ and $T_B$ (in years) and $k_A\left(y{r}^{-1}\right)$ and $k_B\left(ato{m}^{-1}yea{r}^{-1}\right)$.If half-life periods are equal, disintegration rate at the start of disintegration would be:

• A. 0.693
• B. $k_A{T}_A$
• C. both (a) and (b)
• D. None of these

Answer 1

Answer: (C)

both (a) and (b)

$T_A=\frac{0.693}{k_A}$ by first order rate law (based on unit of $k_A$)
$T_B=\frac{1}{k_{B}a}$ by second order rate law (based on unit of $k_B$)
$T_A=T_B$ (given)
$\therefore \frac{0.693}{k_A}=\frac{1}{k_{B}a}$ (a = initial activity)
$\therefore \frac{k_A}{k_B}=0.693a$
$(Rate{)}_A=k_A(a)$
$(Rate{)}_B=k_B(a{)}^2$
$\therefore \frac{(Rate{)}_A}{(Rate{)}_B}=\frac{k_A}{k_{B}a}=0.693=k_A{T}_A$