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

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Solution

Q.
$r = k [A]$ for a reaction. 50% of A is decomposed in 120 minutes. The time taken for 90% decomposition of A is ______ minutes. [2024]

Ans.
(399)

From rate expression it is clear that the reaction is a first order reaction. As 50% of the reactants decompose in 120 min, $t_{1 / 2} = 120$ min. For first order reactions, the rate constant $k$ is given by:   $k = \frac{2.303}{t} \log \frac{a_{o}}{a_{t}}$

  When reaction is 90% complete, 10% reactants are left i.e. $a_{t} = \frac{10}{100} a_{o}, \frac{a_{o}}{a_{t}} = 10$

   $\Rightarrow k = \frac{2.303}{t_{90 %}} \log 10$ (I)

When reaction is 50% complete, 50% reactants are left i.e.   $a_{t} = \frac{50}{100} a_{o}, \frac{a_{o}}{a_{t}} = 2$

   $\Rightarrow k = \frac{2.303}{t_{50 %}} \log 2$ (II)

  Equating I and II

   $\frac{2.303}{t_{90 %}} \log 10 = \frac{2.303}{t_{50 %}} \log 2$

   $\frac{1}{t_{90 %}} \times 1 = \frac{1}{120} \times 0.301$

$t_{90 %} = 398.67 min \approx 399 min$

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