Time required for 99.9 % completion of a first order reaction is _________ times the time required for completion of 90 % reaction.(nearest integer)

Solution

Chemistry ma’am · 3 steps

For first order, time depends only on how much is left: compare 1/1000 left with 1/10 left.

1. First order time. For a first order reaction the time to fall from [A]₀ to [A] is t = 2.303/k log[A]₀/[A]. Only the ratio of starting to remaining amount matters.
2. Fraction left. At 99.9% completion, 0.1% is left, so [A]₀/[A] = 1000. At 90% completion, 10% is left, so [A]₀/[A] = 10.

   t_(99.9%)/t_(90%) = (2.303/k log 1000)/(2.303/k log 10) = 3/1 = 3

3. So. The 2.303/k cancels, and the ratio is log 1000 ÷ log 10 = 3. The answer is 3.

The common slip: 1. Putting the percent completed, 99.9 and 90, into the log instead of the fraction left gives a ratio of about 1, not 3.

The idea to keep. For first order, t ∝ log[A]₀/[A]. Turn percent completion into the fraction left, and the ratio of times is the ratio of those logs.

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