r=k[A] for a reaction, 50 % of A is decomposed in 120 minutes. The time taken for 90 % decomposition of A is _________ minutes.

Solution

Chemistry ma’am · 4 steps

First order, so time depends on the log of the fraction left, not on the amount used up.

1. Spot the order. r = k[A] means the reaction is first order. For a first-order reaction, the time taken depends only on the fraction of A remaining.

   t = 2.303/k log[A]₀/[A]

2. Use the half-life. 50% decomposed in 120 min, so t_(1/2) = 120 min and k = 0.693/120 min⁻¹.
3. Ninety percent gone. 90% decomposed leaves 10% of A, so [A]₀/[A] = 10 and log 10 = 1. Then t₉₀ = 2.303/k = (2.303 × 120)/0.693 = 276.36/0.693 = 398.8 min.
4. So. The time for 90% decomposition is about 399 minutes, roughly 3.32 half-lives.

The common slip: 216. Scaling the time in a straight line, 90/50 × 120, gives 216. That works only for zero order, and here the rate depends on [A].

The idea to keep. For first order, t₉₀ = (log 10)/(log 2) × t_(1/2) ≈ 3.32 t_(1/2). Always put the fraction remaining in the log, never the fraction used up.

Answer from the NTA final answer key. Solution by PrepWiser.