The rate of first-order reaction is 0.04 mol at 10 seconds and 0.03 mol at 20 seconds after initiation of the reaction. The half-life period of the reaction is [2016]

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Solution

Q.
The rate of first-order reaction is 0.04 mol $L^{- 1}$ $s^{- 1}$ at 10 seconds and 0.03 mol $L^{- 1}$ $s^{- 1}$ at 20 seconds after initiation of the reaction. The half-life period of the reaction is [2016]

1 44.1 s

2 54.1 s

3 24.1 s

4 34.1 s

Ans.
(3)

For a first order reaction, A $\to$ Products and for concentration of the reactant at two different times,

$k = \frac{2.303}{t_{2} - t_{1}} \log \frac{{[A]}_{1}}{{[A]}_{2}}$

$∴$ $k = \frac{2.303}{t_{2} - t_{1}} \log \frac{{(rate)}_{1}}{{(rate)}_{2}}$ $(∵ rate \propto [A])$

$k = \frac{2.303}{(20 - 10)} \log (\frac{0.04}{0.03}) = 0.0287$ ${sec}^{- 1}$

$t_{1 / 2} = \frac{0.693}{k} = \frac{0.693}{0.0287 {sec}^{- 1}} = 24.14$ $sec$

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