A and B decompose by first-order kinetics with half-lives of 54.0 min and 18.0 min respectively. Starting from equimolar amounts of A and B, find the time (in min) at which [A] = 16[B].

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EEJ MAIN Chemistry QUESTION #5539

Question 1

A and B decompose by first-order kinetics with half-lives of 54.0 min and 18.0 min respectively. Starting from equimolar amounts of A and B, find the time (in min) at which $[A] = 16[B]$.

54
108✔️
162
216

Correct Answer Explanation

For equimolar start, $[A]_0 = [B]_0 = C$. $[A] = C\left(\frac{1}{2}\right)^{t/54}$, $[B] = C\left(\frac{1}{2}\right)^{t/18}$. Setting $[A]=16[B]$: $\left(\frac{1}{2}\right)^{t/54} = 16\left(\frac{1}{2}\right)^{t/18}$. $2^{-t/54} = 2^4 \cdot 2^{-t/18}$. $-\frac{t}{54} = 4 - \frac{t}{18}$. $\frac{t}{18}-\frac{t}{54} = 4 \Rightarrow \frac{3t-t}{54}=4 \Rightarrow \frac{2t}{54}=4 \Rightarrow t = 108\ \text{min}$.

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