r = k[A]$r = k[A]$ for a reaction, 50\%$50\%$ of A is decomposed in 120text minutes$120\text{ minutes}$. The time taken for 90\%$90\%$ decomposition of A is ________ minutes.

### Related Formula k = frac0.693t_1/2$$k = \frac{0.693}{t_{1/2}}$$ t = frac2.303k logleft( fracaa - x right)$$t = \frac{2.303}{k} \log\left( \frac{a}{a - x} \right)$$ ### Core Logic Since r = k[A]$r = k[A]$, the reaction follows first-order kinetics. The half-life (50\%$50\%$ decomposition) is t_1/2 = 120text minutes$t_{1/2} = 120\text{ minutes}$. ### Step 1: Calculating for 90% decomposition For 90\%$90\%$ completion of the reaction, [A]_0 = 100$[A]_0 = 100$ and [A]_t = 100 - 90 = 10$[A]_t = 100 - 90 = 10$. t = frac2.303k log frac10010$$t = \frac{2.303}{k} \log \frac{100}{10}$$ t = frac2.303left( frac0.693t_1/2 right) log (10)$$t = \frac{2.303}{\left( \frac{0.693}{t_{1/2}} \right)} \log (10)$$ t = frac2.303 times 1200.693 times 1$$t = \frac{2.303 \times 120}{0.693} \times 1$$ t = 398.78text minutes$$t = 398.78\text{ minutes}$$ Rounding off to the nearest integer, we get 399text minutes$399\text{ minutes}$. ### Pattern Recognition For a first order reaction, t_90\% approx 3.32 times t_50\%$t_{90\%} \approx 3.32 \times t_{50\%}$. 120 times 3.32 = 398.4$120 \times 3.32 = 398.4$, so roughly 399$399$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Chemistry: Chemical Kinetics