Question

A flask contains a mixture of two compounds AB and XY. Both of these decompose on heating by first order reactions. If the half-life of AB is 30 min and that of XY is 10 min, how long will it take for the concentration of AB to be four times that of XY? Assume the initial concentration of both AB and XY to be the same.

• A. 30 min
• B. 40 min
• C. 50 min
• D. 60 min

Answer 1

The correct option is A 30 min

$ln\frac{a_0}{a}=kt$ for a first order reaction.
Let $t_1$ and $t_2$ be half lives of AB and XY respectively.
$k_1=\frac{ln2}{t_1},k_2=\frac{ln2}{t_2}$
$ln\frac{a_0}{a_{AB}}=\frac{ln2}{30}t$
$ln\frac{a_0}{a_{XY}}=\frac{ln2}{10}t$
And $a_{AB}=4a_{XY}$
$ln\frac{a_0}{4a_{XY}}=\frac{ln2}{30}t$
Removing logs (taking exponents on LHS and RHS):
$2^{\frac{t}{10}-\frac{t}{30}}=2^2$
$\frac{t}{10}-\frac{t}{30}=2$
$20t=600$
$t=30\text{min}$