For a given parallel reaction if both the reaction follows 1^st order kinetics
If initially 10 moles A is taken and $\frac{K_{1}}{K_{2}}$ is 0.5 then select the correct statement(s).

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At any instant the ratio of [B] to [C] is constant and equal to $\frac{1}{3}$

At any instant the ratio of [B] to [C] is constant and equal to $\frac{1}{2}$

At any instant when no of moles of B formed is 2, then no of moles of C formed at that instant is equal to 4

At any instant when no of moles of B formed is 2, then no of moles of A remaining at that instant is equal to 7

answer is B, C.

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Detailed Solution

At any instant
$\begin{array}{l} \frac{[B]}{[C]} = \frac{2 k_{1}}{3 k_{2}} = \frac{2}{3} \times \frac{1}{2} = \frac{1}{3} and [A] + \frac{[B]}{2} + \frac{[C]}{3} = [A]_{0} \\ ∵ n_{A} + \frac{n_{B}}{2} + \frac{n_{C}}{3} = {(n_{A})}_{i} \\ \frac{n_{B}}{n_{C}} = \frac{1}{3} \end{array}$
$∴$ when $n_{B} = 2$
$\begin{aligned} n_{C} = 6 \\ n_{A} = 10 - 1 - 2 = 7 \end{aligned}$